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Properties

Label 2-352-11.3-c1-0-0
Degree 22
Conductor 352352
Sign −0.919−0.392i-0.919 - 0.392i
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  + (0.945 + 2.90i)3-s + (−2.43 − 1.76i)5-s + (−0.483 + 1.48i)7-s + (−5.14 + 3.73i)9-s + (−3.21 + 0.815i)11-s + (−2.95 + 2.14i)13-s + (2.84 − 8.74i)15-s + (3.62 + 2.63i)17-s + (0.848 + 2.61i)19-s − 4.78·21-s + 4.77·23-s + (1.24 + 3.83i)25-s + (−8.31 − 6.03i)27-s + (1.53 − 4.72i)29-s + (−0.394 + 0.286i)31-s + ⋯
L(s)  = 1  + (0.545 + 1.67i)3-s + (−1.08 − 0.790i)5-s + (−0.182 + 0.562i)7-s + (−1.71 + 1.24i)9-s + (−0.969 + 0.245i)11-s + (−0.820 + 0.596i)13-s + (0.733 − 2.25i)15-s + (0.878 + 0.638i)17-s + (0.194 + 0.599i)19-s − 1.04·21-s + 0.995·23-s + (0.249 + 0.767i)25-s + (−1.59 − 1.16i)27-s + (0.285 − 0.877i)29-s + (−0.0708 + 0.0514i)31-s + ⋯

Functional equation

Λ(s)=(352s/2 ΓC(s) L(s)=((−0.919−0.392i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 - 0.392i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(352s/2 ΓC(s+1/2) L(s)=((−0.919−0.392i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.919 - 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 352352    =    25⋅112^{5} \cdot 11
Sign: −0.919−0.392i-0.919 - 0.392i
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), −0.919−0.392i)(2,\ 352,\ (\ :1/2),\ -0.919 - 0.392i)

Particular Values

L(1)L(1) ≈\approx 0.185773+0.908114i0.185773 + 0.908114i
L(12)L(\frac12) ≈\approx 0.185773+0.908114i0.185773 + 0.908114i
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+(3.21−0.815i)T 1 + (3.21 - 0.815i)T
good3 1+(−0.945−2.90i)T+(−2.42+1.76i)T2 1 + (-0.945 - 2.90i)T + (-2.42 + 1.76i)T^{2}
5 1+(2.43+1.76i)T+(1.54+4.75i)T2 1 + (2.43 + 1.76i)T + (1.54 + 4.75i)T^{2}
7 1+(0.483−1.48i)T+(−5.66−4.11i)T2 1 + (0.483 - 1.48i)T + (-5.66 - 4.11i)T^{2}
13 1+(2.95−2.14i)T+(4.01−12.3i)T2 1 + (2.95 - 2.14i)T + (4.01 - 12.3i)T^{2}
17 1+(−3.62−2.63i)T+(5.25+16.1i)T2 1 + (-3.62 - 2.63i)T + (5.25 + 16.1i)T^{2}
19 1+(−0.848−2.61i)T+(−15.3+11.1i)T2 1 + (-0.848 - 2.61i)T + (-15.3 + 11.1i)T^{2}
23 1−4.77T+23T2 1 - 4.77T + 23T^{2}
29 1+(−1.53+4.72i)T+(−23.4−17.0i)T2 1 + (-1.53 + 4.72i)T + (-23.4 - 17.0i)T^{2}
31 1+(0.394−0.286i)T+(9.57−29.4i)T2 1 + (0.394 - 0.286i)T + (9.57 - 29.4i)T^{2}
37 1+(3.28−10.1i)T+(−29.9−21.7i)T2 1 + (3.28 - 10.1i)T + (-29.9 - 21.7i)T^{2}
41 1+(3.87+11.9i)T+(−33.1+24.0i)T2 1 + (3.87 + 11.9i)T + (-33.1 + 24.0i)T^{2}
43 1−7.45T+43T2 1 - 7.45T + 43T^{2}
47 1+(−3.10−9.54i)T+(−38.0+27.6i)T2 1 + (-3.10 - 9.54i)T + (-38.0 + 27.6i)T^{2}
53 1+(−4.26+3.09i)T+(16.3−50.4i)T2 1 + (-4.26 + 3.09i)T + (16.3 - 50.4i)T^{2}
59 1+(2.76−8.49i)T+(−47.7−34.6i)T2 1 + (2.76 - 8.49i)T + (-47.7 - 34.6i)T^{2}
61 1+(1.75+1.27i)T+(18.8+58.0i)T2 1 + (1.75 + 1.27i)T + (18.8 + 58.0i)T^{2}
67 1−0.709T+67T2 1 - 0.709T + 67T^{2}
71 1+(0.654+0.475i)T+(21.9+67.5i)T2 1 + (0.654 + 0.475i)T + (21.9 + 67.5i)T^{2}
73 1+(0.163−0.502i)T+(−59.0−42.9i)T2 1 + (0.163 - 0.502i)T + (-59.0 - 42.9i)T^{2}
79 1+(4.70−3.41i)T+(24.4−75.1i)T2 1 + (4.70 - 3.41i)T + (24.4 - 75.1i)T^{2}
83 1+(−5.73−4.16i)T+(25.6+78.9i)T2 1 + (-5.73 - 4.16i)T + (25.6 + 78.9i)T^{2}
89 1−7.76T+89T2 1 - 7.76T + 89T^{2}
97 1+(2.93−2.13i)T+(29.9−92.2i)T2 1 + (2.93 - 2.13i)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

−11.92016515906435327071233931897, −10.72942442293805259369545520615, −9.982043797460385382268485366982, −9.114619781834648795657080817256, −8.376791018179645007649965376078, −7.56403208915759376304362732933, −5.55347805948240643430389816208, −4.74766686520069704178673566643, −3.92152292168912939128929011580, −2.77445724219536714516702356418, 0.58808577891943064826750689529, 2.65409112385666367686263045580, 3.35387390102636686640935641752, 5.28149470208903270769948538875, 6.77525290834504355681901517322, 7.48730917342013462764212904235, 7.70383171421005629619407561026, 8.899045313641513277680394241720, 10.35164235385366722893354734150, 11.26065473675989049486360364809

Graph of the ZZ-function along the critical line