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Properties

Label 2-378-27.13-c1-0-8
Degree 22
Conductor 378378
Sign 0.993+0.116i0.993 + 0.116i
Analytic cond. 3.018343.01834
Root an. cond. 1.737331.73733
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.173 + 0.984i)2-s + (−1.70 + 0.300i)3-s + (−0.939 + 0.342i)4-s + (−1.03 − 0.866i)5-s + (−0.592 − 1.62i)6-s + (−0.939 − 0.342i)7-s + (−0.5 − 0.866i)8-s + (2.81 − 1.02i)9-s + (0.673 − 1.16i)10-s + (3.64 − 3.05i)11-s + (1.49 − 0.866i)12-s + (0.266 − 1.50i)13-s + (0.173 − 0.984i)14-s + (2.02 + 1.16i)15-s + (0.766 − 0.642i)16-s + (−1.11 + 1.92i)17-s + ⋯
L(s)  = 1  + (0.122 + 0.696i)2-s + (−0.984 + 0.173i)3-s + (−0.469 + 0.171i)4-s + (−0.461 − 0.387i)5-s + (−0.241 − 0.664i)6-s + (−0.355 − 0.129i)7-s + (−0.176 − 0.306i)8-s + (0.939 − 0.342i)9-s + (0.213 − 0.368i)10-s + (1.09 − 0.922i)11-s + (0.433 − 0.249i)12-s + (0.0737 − 0.418i)13-s + (0.0464 − 0.263i)14-s + (0.521 + 0.301i)15-s + (0.191 − 0.160i)16-s + (−0.270 + 0.467i)17-s + ⋯

Functional equation

Λ(s)=(378s/2 ΓC(s) L(s)=((0.993+0.116i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(378s/2 ΓC(s+1/2) L(s)=((0.993+0.116i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 378378    =    2⋅33⋅72 \cdot 3^{3} \cdot 7
Sign: 0.993+0.116i0.993 + 0.116i
Analytic conductor: 3.018343.01834
Root analytic conductor: 1.737331.73733
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ378(337,⋅)\chi_{378} (337, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 378, ( :1/2), 0.993+0.116i)(2,\ 378,\ (\ :1/2),\ 0.993 + 0.116i)

Particular Values

L(1)L(1) ≈\approx 0.871721−0.0507719i0.871721 - 0.0507719i
L(12)L(\frac12) ≈\approx 0.871721−0.0507719i0.871721 - 0.0507719i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(−0.173−0.984i)T 1 + (-0.173 - 0.984i)T
3 1+(1.70−0.300i)T 1 + (1.70 - 0.300i)T
7 1+(0.939+0.342i)T 1 + (0.939 + 0.342i)T
good5 1+(1.03+0.866i)T+(0.868+4.92i)T2 1 + (1.03 + 0.866i)T + (0.868 + 4.92i)T^{2}
11 1+(−3.64+3.05i)T+(1.91−10.8i)T2 1 + (-3.64 + 3.05i)T + (1.91 - 10.8i)T^{2}
13 1+(−0.266+1.50i)T+(−12.2−4.44i)T2 1 + (-0.266 + 1.50i)T + (-12.2 - 4.44i)T^{2}
17 1+(1.11−1.92i)T+(−8.5−14.7i)T2 1 + (1.11 - 1.92i)T + (-8.5 - 14.7i)T^{2}
19 1+(−2.79−4.84i)T+(−9.5+16.4i)T2 1 + (-2.79 - 4.84i)T + (-9.5 + 16.4i)T^{2}
23 1+(−7.57+2.75i)T+(17.6−14.7i)T2 1 + (-7.57 + 2.75i)T + (17.6 - 14.7i)T^{2}
29 1+(1.85+10.5i)T+(−27.2+9.91i)T2 1 + (1.85 + 10.5i)T + (-27.2 + 9.91i)T^{2}
31 1+(−1.37+0.502i)T+(23.7−19.9i)T2 1 + (-1.37 + 0.502i)T + (23.7 - 19.9i)T^{2}
37 1+(−0.815+1.41i)T+(−18.5−32.0i)T2 1 + (-0.815 + 1.41i)T + (-18.5 - 32.0i)T^{2}
41 1+(−1.75+9.97i)T+(−38.5−14.0i)T2 1 + (-1.75 + 9.97i)T + (-38.5 - 14.0i)T^{2}
43 1+(6.70−5.63i)T+(7.46−42.3i)T2 1 + (6.70 - 5.63i)T + (7.46 - 42.3i)T^{2}
47 1+(−8.35−3.03i)T+(36.0+30.2i)T2 1 + (-8.35 - 3.03i)T + (36.0 + 30.2i)T^{2}
53 1−1.32T+53T2 1 - 1.32T + 53T^{2}
59 1+(7.16+6.01i)T+(10.2+58.1i)T2 1 + (7.16 + 6.01i)T + (10.2 + 58.1i)T^{2}
61 1+(−3.08−1.12i)T+(46.7+39.2i)T2 1 + (-3.08 - 1.12i)T + (46.7 + 39.2i)T^{2}
67 1+(0.470−2.66i)T+(−62.9−22.9i)T2 1 + (0.470 - 2.66i)T + (-62.9 - 22.9i)T^{2}
71 1+(2.10−3.64i)T+(−35.5−61.4i)T2 1 + (2.10 - 3.64i)T + (-35.5 - 61.4i)T^{2}
73 1+(4.54+7.87i)T+(−36.5+63.2i)T2 1 + (4.54 + 7.87i)T + (-36.5 + 63.2i)T^{2}
79 1+(−0.556−3.15i)T+(−74.2+27.0i)T2 1 + (-0.556 - 3.15i)T + (-74.2 + 27.0i)T^{2}
83 1+(2.22+12.6i)T+(−77.9+28.3i)T2 1 + (2.22 + 12.6i)T + (-77.9 + 28.3i)T^{2}
89 1+(0.779+1.35i)T+(−44.5+77.0i)T2 1 + (0.779 + 1.35i)T + (-44.5 + 77.0i)T^{2}
97 1+(9.91−8.31i)T+(16.8−95.5i)T2 1 + (9.91 - 8.31i)T + (16.8 - 95.5i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−11.47620145947200787826983742815, −10.47502985172644888923077277655, −9.454610451456002526289722313124, −8.496728257358985066141139419258, −7.44727101291300540171248202030, −6.33094094610710624952731851290, −5.77420506695725453505405418021, −4.48056608614308113383340474784, −3.63334277362969873672709770248, −0.75551355626911789136133263968, 1.36587140397128264326838243014, 3.16563681769817311945100985196, 4.44483479195359510960871454432, 5.33058048778540071897575458397, 6.89997068848061516252607600840, 7.10996443877423341199700773964, 9.028039381733479485965923485953, 9.624311882683919499542113070624, 10.82561694135699951935131934885, 11.42910366964784561219462478145

Graph of the ZZ-function along the critical line