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Properties

Label 2-352-11.3-c1-0-7
Degree 22
Conductor 352352
Sign 0.212+0.977i0.212 + 0.977i
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  + (−0.780 − 2.40i)3-s + (2.30 + 1.67i)5-s + (0.780 − 2.40i)7-s + (−2.73 + 1.98i)9-s + (2.52 + 2.14i)11-s + (3.92 − 2.85i)13-s + (2.22 − 6.85i)15-s + (−4.54 − 3.30i)17-s + (0.780 + 2.40i)19-s − 6.38·21-s − 8.17·23-s + (0.972 + 2.99i)25-s + (0.780 + 0.567i)27-s + (0.427 − 1.31i)29-s + (5.35 − 3.88i)31-s + ⋯
L(s)  = 1  + (−0.450 − 1.38i)3-s + (1.03 + 0.750i)5-s + (0.295 − 0.908i)7-s + (−0.912 + 0.662i)9-s + (0.761 + 0.647i)11-s + (1.08 − 0.791i)13-s + (0.575 − 1.77i)15-s + (−1.10 − 0.800i)17-s + (0.179 + 0.551i)19-s − 1.39·21-s − 1.70·23-s + (0.194 + 0.598i)25-s + (0.150 + 0.109i)27-s + (0.0793 − 0.244i)29-s + (0.961 − 0.698i)31-s + ⋯

Functional equation

Λ(s)=(352s/2 ΓC(s) L(s)=((0.212+0.977i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.212 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(352s/2 ΓC(s+1/2) L(s)=((0.212+0.977i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.212 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 352352    =    25⋅112^{5} \cdot 11
Sign: 0.212+0.977i0.212 + 0.977i
Analytic conductor: 2.810732.81073
Root analytic conductor: 1.676521.67652
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ352(289,⋅)\chi_{352} (289, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 352, ( :1/2), 0.212+0.977i)(2,\ 352,\ (\ :1/2),\ 0.212 + 0.977i)

Particular Values

L(1)L(1) ≈\approx 1.11090−0.895564i1.11090 - 0.895564i
L(12)L(\frac12) ≈\approx 1.11090−0.895564i1.11090 - 0.895564i
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+(−2.52−2.14i)T 1 + (-2.52 - 2.14i)T
good3 1+(0.780+2.40i)T+(−2.42+1.76i)T2 1 + (0.780 + 2.40i)T + (-2.42 + 1.76i)T^{2}
5 1+(−2.30−1.67i)T+(1.54+4.75i)T2 1 + (-2.30 - 1.67i)T + (1.54 + 4.75i)T^{2}
7 1+(−0.780+2.40i)T+(−5.66−4.11i)T2 1 + (-0.780 + 2.40i)T + (-5.66 - 4.11i)T^{2}
13 1+(−3.92+2.85i)T+(4.01−12.3i)T2 1 + (-3.92 + 2.85i)T + (4.01 - 12.3i)T^{2}
17 1+(4.54+3.30i)T+(5.25+16.1i)T2 1 + (4.54 + 3.30i)T + (5.25 + 16.1i)T^{2}
19 1+(−0.780−2.40i)T+(−15.3+11.1i)T2 1 + (-0.780 - 2.40i)T + (-15.3 + 11.1i)T^{2}
23 1+8.17T+23T2 1 + 8.17T + 23T^{2}
29 1+(−0.427+1.31i)T+(−23.4−17.0i)T2 1 + (-0.427 + 1.31i)T + (-23.4 - 17.0i)T^{2}
31 1+(−5.35+3.88i)T+(9.57−29.4i)T2 1 + (-5.35 + 3.88i)T + (9.57 - 29.4i)T^{2}
37 1+(0.190−0.587i)T+(−29.9−21.7i)T2 1 + (0.190 - 0.587i)T + (-29.9 - 21.7i)T^{2}
41 1+(−2.66−8.19i)T+(−33.1+24.0i)T2 1 + (-2.66 - 8.19i)T + (-33.1 + 24.0i)T^{2}
43 1−3.12T+43T2 1 - 3.12T + 43T^{2}
47 1+(0.184+0.567i)T+(−38.0+27.6i)T2 1 + (0.184 + 0.567i)T + (-38.0 + 27.6i)T^{2}
53 1+(0.309−0.224i)T+(16.3−50.4i)T2 1 + (0.309 - 0.224i)T + (16.3 - 50.4i)T^{2}
59 1+(3.30−10.1i)T+(−47.7−34.6i)T2 1 + (3.30 - 10.1i)T + (-47.7 - 34.6i)T^{2}
61 1+(−2.92−2.12i)T+(18.8+58.0i)T2 1 + (-2.92 - 2.12i)T + (18.8 + 58.0i)T^{2}
67 1+13.2T+67T2 1 + 13.2T + 67T^{2}
71 1+(−9.43−6.85i)T+(21.9+67.5i)T2 1 + (-9.43 - 6.85i)T + (21.9 + 67.5i)T^{2}
73 1+(−1.33+4.11i)T+(−59.0−42.9i)T2 1 + (-1.33 + 4.11i)T + (-59.0 - 42.9i)T^{2}
79 1+(2.82−2.05i)T+(24.4−75.1i)T2 1 + (2.82 - 2.05i)T + (24.4 - 75.1i)T^{2}
83 1+(−9.43−6.85i)T+(25.6+78.9i)T2 1 + (-9.43 - 6.85i)T + (25.6 + 78.9i)T^{2}
89 1+7.23T+89T2 1 + 7.23T + 89T^{2}
97 1+(14.3−10.4i)T+(29.9−92.2i)T2 1 + (14.3 - 10.4i)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

−11.36488693847930734768703570863, −10.45323544454541444895031903361, −9.627876680177328023973410376934, −8.142185277465513659326682211584, −7.30011166184464454181512614819, −6.39793106243460855434006565220, −5.95396009283343828859561685685, −4.19183158978723017204706494482, −2.39249358730978230710493289303, −1.21618799661730450543885987417, 1.87426610640449237088014709512, 3.82044621435096490803343376403, 4.73912851782466611394473301457, 5.79389771833007636924653205474, 6.30934604862076730794952590361, 8.616325231783190068732858451917, 8.921612263948890093283339486247, 9.734850362993715757541245998700, 10.76219978367779286457532929348, 11.46449705451867148243816212207

Graph of the ZZ-function along the critical line