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Properties

Label 2-352-44.35-c1-0-5
Degree 22
Conductor 352352
Sign 0.928+0.371i0.928 + 0.371i
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  + (−0.708 − 0.975i)3-s + (0.671 + 2.06i)5-s + (−1.18 − 0.858i)7-s + (0.477 − 1.47i)9-s + (3.29 − 0.371i)11-s + (5.50 + 1.78i)13-s + (1.54 − 2.11i)15-s + (−2.83 + 0.921i)17-s + (4.63 − 3.36i)19-s + 1.76i·21-s − 0.400i·23-s + (0.222 − 0.162i)25-s + (−5.21 + 1.69i)27-s + (2.26 − 3.11i)29-s + (1.41 + 0.460i)31-s + ⋯
L(s)  = 1  + (−0.409 − 0.563i)3-s + (0.300 + 0.924i)5-s + (−0.446 − 0.324i)7-s + (0.159 − 0.490i)9-s + (0.993 − 0.112i)11-s + (1.52 + 0.496i)13-s + (0.397 − 0.547i)15-s + (−0.688 + 0.223i)17-s + (1.06 − 0.773i)19-s + 0.384i·21-s − 0.0834i·23-s + (0.0445 − 0.0324i)25-s + (−1.00 + 0.325i)27-s + (0.419 − 0.577i)29-s + (0.254 + 0.0827i)31-s + ⋯

Functional equation

Λ(s)=(352s/2 ΓC(s) L(s)=((0.928+0.371i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(352s/2 ΓC(s+1/2) L(s)=((0.928+0.371i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 352352    =    25⋅112^{5} \cdot 11
Sign: 0.928+0.371i0.928 + 0.371i
Analytic conductor: 2.810732.81073
Root analytic conductor: 1.676521.67652
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ352(255,⋅)\chi_{352} (255, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 352, ( :1/2), 0.928+0.371i)(2,\ 352,\ (\ :1/2),\ 0.928 + 0.371i)

Particular Values

L(1)L(1) ≈\approx 1.28397−0.247514i1.28397 - 0.247514i
L(12)L(\frac12) ≈\approx 1.28397−0.247514i1.28397 - 0.247514i
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+(−3.29+0.371i)T 1 + (-3.29 + 0.371i)T
good3 1+(0.708+0.975i)T+(−0.927+2.85i)T2 1 + (0.708 + 0.975i)T + (-0.927 + 2.85i)T^{2}
5 1+(−0.671−2.06i)T+(−4.04+2.93i)T2 1 + (-0.671 - 2.06i)T + (-4.04 + 2.93i)T^{2}
7 1+(1.18+0.858i)T+(2.16+6.65i)T2 1 + (1.18 + 0.858i)T + (2.16 + 6.65i)T^{2}
13 1+(−5.50−1.78i)T+(10.5+7.64i)T2 1 + (-5.50 - 1.78i)T + (10.5 + 7.64i)T^{2}
17 1+(2.83−0.921i)T+(13.7−9.99i)T2 1 + (2.83 - 0.921i)T + (13.7 - 9.99i)T^{2}
19 1+(−4.63+3.36i)T+(5.87−18.0i)T2 1 + (-4.63 + 3.36i)T + (5.87 - 18.0i)T^{2}
23 1+0.400iT−23T2 1 + 0.400iT - 23T^{2}
29 1+(−2.26+3.11i)T+(−8.96−27.5i)T2 1 + (-2.26 + 3.11i)T + (-8.96 - 27.5i)T^{2}
31 1+(−1.41−0.460i)T+(25.0+18.2i)T2 1 + (-1.41 - 0.460i)T + (25.0 + 18.2i)T^{2}
37 1+(−5.01−3.64i)T+(11.4+35.1i)T2 1 + (-5.01 - 3.64i)T + (11.4 + 35.1i)T^{2}
41 1+(2.50+3.45i)T+(−12.6+38.9i)T2 1 + (2.50 + 3.45i)T + (-12.6 + 38.9i)T^{2}
43 1+6.17T+43T2 1 + 6.17T + 43T^{2}
47 1+(−4.79−6.59i)T+(−14.5+44.6i)T2 1 + (-4.79 - 6.59i)T + (-14.5 + 44.6i)T^{2}
53 1+(0.633−1.95i)T+(−42.8−31.1i)T2 1 + (0.633 - 1.95i)T + (-42.8 - 31.1i)T^{2}
59 1+(7.19−9.90i)T+(−18.2−56.1i)T2 1 + (7.19 - 9.90i)T + (-18.2 - 56.1i)T^{2}
61 1+(12.4−4.04i)T+(49.3−35.8i)T2 1 + (12.4 - 4.04i)T + (49.3 - 35.8i)T^{2}
67 1−2.72iT−67T2 1 - 2.72iT - 67T^{2}
71 1+(9.62−3.12i)T+(57.4−41.7i)T2 1 + (9.62 - 3.12i)T + (57.4 - 41.7i)T^{2}
73 1+(−2.77+3.81i)T+(−22.5−69.4i)T2 1 + (-2.77 + 3.81i)T + (-22.5 - 69.4i)T^{2}
79 1+(−2.34+7.20i)T+(−63.9−46.4i)T2 1 + (-2.34 + 7.20i)T + (-63.9 - 46.4i)T^{2}
83 1+(−3.31−10.1i)T+(−67.1+48.7i)T2 1 + (-3.31 - 10.1i)T + (-67.1 + 48.7i)T^{2}
89 1−1.16T+89T2 1 - 1.16T + 89T^{2}
97 1+(−5.15+15.8i)T+(−78.4−57.0i)T2 1 + (-5.15 + 15.8i)T + (-78.4 - 57.0i)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

−11.44580862333842496688687355265, −10.67941624305425089404377904894, −9.575294321717000810405467587978, −8.756062509258203765013011657132, −7.25986730883858073152639028279, −6.48022865854748868141812826718, −6.12428658234786361521634115648, −4.20446104662802386995652678060, −3.09622701487837386607545644637, −1.25649251891578657702648337296, 1.42263940345758538707034452194, 3.46953802390108562422197242324, 4.63911690629511071425429934332, 5.59247162323690720421559975825, 6.46382918777164106649177604987, 7.955572447432593493982841242285, 8.962213505370088266384374181336, 9.587882318764931725183754004772, 10.63226564925529209691446934824, 11.47493703526182201529154926437

Graph of the ZZ-function along the critical line