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Properties

Label 2-352-44.7-c1-0-8
Degree 22
Conductor 352352
Sign 0.998−0.0505i0.998 - 0.0505i
Analytic cond. 2.810732.81073
Root an. cond. 1.676521.67652
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  + (2.56 + 0.832i)3-s + (1.18 − 0.862i)5-s + (−0.845 − 2.60i)7-s + (3.45 + 2.50i)9-s + (−0.122 + 3.31i)11-s + (1.50 − 2.06i)13-s + (3.76 − 1.22i)15-s + (−3.48 − 4.79i)17-s + (−0.531 + 1.63i)19-s − 7.37i·21-s + 7.75i·23-s + (−0.880 + 2.70i)25-s + (2.00 + 2.76i)27-s + (−3.07 + 0.997i)29-s + (−3.93 + 5.42i)31-s + ⋯
L(s)  = 1  + (1.48 + 0.480i)3-s + (0.530 − 0.385i)5-s + (−0.319 − 0.983i)7-s + (1.15 + 0.835i)9-s + (−0.0369 + 0.999i)11-s + (0.417 − 0.574i)13-s + (0.971 − 0.315i)15-s + (−0.844 − 1.16i)17-s + (−0.121 + 0.375i)19-s − 1.60i·21-s + 1.61i·23-s + (−0.176 + 0.541i)25-s + (0.385 + 0.531i)27-s + (−0.570 + 0.185i)29-s + (−0.707 + 0.973i)31-s + ⋯

Functional equation

Λ(s)=(352s/2 ΓC(s) L(s)=((0.998−0.0505i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0505i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(352s/2 ΓC(s+1/2) L(s)=((0.998−0.0505i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0505i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 352352    =    25⋅112^{5} \cdot 11
Sign: 0.998−0.0505i0.998 - 0.0505i
Analytic conductor: 2.810732.81073
Root analytic conductor: 1.676521.67652
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ352(95,⋅)\chi_{352} (95, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 352, ( :1/2), 0.998−0.0505i)(2,\ 352,\ (\ :1/2),\ 0.998 - 0.0505i)

Particular Values

L(1)L(1) ≈\approx 2.16548+0.0547556i2.16548 + 0.0547556i
L(12)L(\frac12) ≈\approx 2.16548+0.0547556i2.16548 + 0.0547556i
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 1
11 1+(0.122−3.31i)T 1 + (0.122 - 3.31i)T
good3 1+(−2.56−0.832i)T+(2.42+1.76i)T2 1 + (-2.56 - 0.832i)T + (2.42 + 1.76i)T^{2}
5 1+(−1.18+0.862i)T+(1.54−4.75i)T2 1 + (-1.18 + 0.862i)T + (1.54 - 4.75i)T^{2}
7 1+(0.845+2.60i)T+(−5.66+4.11i)T2 1 + (0.845 + 2.60i)T + (-5.66 + 4.11i)T^{2}
13 1+(−1.50+2.06i)T+(−4.01−12.3i)T2 1 + (-1.50 + 2.06i)T + (-4.01 - 12.3i)T^{2}
17 1+(3.48+4.79i)T+(−5.25+16.1i)T2 1 + (3.48 + 4.79i)T + (-5.25 + 16.1i)T^{2}
19 1+(0.531−1.63i)T+(−15.3−11.1i)T2 1 + (0.531 - 1.63i)T + (-15.3 - 11.1i)T^{2}
23 1−7.75iT−23T2 1 - 7.75iT - 23T^{2}
29 1+(3.07−0.997i)T+(23.4−17.0i)T2 1 + (3.07 - 0.997i)T + (23.4 - 17.0i)T^{2}
31 1+(3.93−5.42i)T+(−9.57−29.4i)T2 1 + (3.93 - 5.42i)T + (-9.57 - 29.4i)T^{2}
37 1+(2.50+7.70i)T+(−29.9+21.7i)T2 1 + (2.50 + 7.70i)T + (-29.9 + 21.7i)T^{2}
41 1+(−10.8−3.52i)T+(33.1+24.0i)T2 1 + (-10.8 - 3.52i)T + (33.1 + 24.0i)T^{2}
43 1−0.0675T+43T2 1 - 0.0675T + 43T^{2}
47 1+(6.48+2.10i)T+(38.0+27.6i)T2 1 + (6.48 + 2.10i)T + (38.0 + 27.6i)T^{2}
53 1+(4.99+3.62i)T+(16.3+50.4i)T2 1 + (4.99 + 3.62i)T + (16.3 + 50.4i)T^{2}
59 1+(−5.13+1.66i)T+(47.7−34.6i)T2 1 + (-5.13 + 1.66i)T + (47.7 - 34.6i)T^{2}
61 1+(−4.35−5.99i)T+(−18.8+58.0i)T2 1 + (-4.35 - 5.99i)T + (-18.8 + 58.0i)T^{2}
67 1+11.2iT−67T2 1 + 11.2iT - 67T^{2}
71 1+(−5.04−6.94i)T+(−21.9+67.5i)T2 1 + (-5.04 - 6.94i)T + (-21.9 + 67.5i)T^{2}
73 1+(15.1−4.92i)T+(59.0−42.9i)T2 1 + (15.1 - 4.92i)T + (59.0 - 42.9i)T^{2}
79 1+(8.56+6.22i)T+(24.4+75.1i)T2 1 + (8.56 + 6.22i)T + (24.4 + 75.1i)T^{2}
83 1+(0.450−0.327i)T+(25.6−78.9i)T2 1 + (0.450 - 0.327i)T + (25.6 - 78.9i)T^{2}
89 1−13.1T+89T2 1 - 13.1T + 89T^{2}
97 1+(0.746+0.542i)T+(29.9+92.2i)T2 1 + (0.746 + 0.542i)T + (29.9 + 92.2i)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

−11.29123055062883657630059725245, −10.20842785846865768830754598967, −9.496450594791890679970039515781, −8.985354631554722030447876899555, −7.70001059263530283707714516799, −7.10573480434801295877987939103, −5.39373115561712873101683586035, −4.17735498953195049301956141062, −3.25836737208530271451225917859, −1.83002677554276318407433452184, 2.05926427926641434155925742212, 2.80904908156968899702677245021, 4.07631121588351568947015203578, 5.98249428484381266271284332054, 6.59109518452063345901097151783, 8.006784860305837168984941955116, 8.749678129023686787125788548062, 9.212708452296337732544739952483, 10.44616462391110388524477588898, 11.46298386423063321190381501479

Graph of the ZZ-function along the critical line