Properties

Label 2-736-184.101-c1-0-16
Degree $2$
Conductor $736$
Sign $0.100 + 0.994i$
Analytic cond. $5.87698$
Root an. cond. $2.42425$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.32 + 2.01i)3-s + (2.15 − 0.309i)5-s + (0.974 − 2.13i)7-s + (0.920 − 6.40i)9-s + (0.0777 − 0.264i)11-s + (−3.89 + 1.77i)13-s + (−4.38 + 5.05i)15-s + (−4.80 − 3.08i)17-s + (−3.39 − 5.28i)19-s + (2.03 + 6.92i)21-s + (−4.77 − 0.395i)23-s + (−0.256 + 0.0753i)25-s + (5.77 + 8.98i)27-s + (1.47 − 2.29i)29-s + (3.50 − 4.04i)31-s + ⋯
L(s)  = 1  + (−1.34 + 1.16i)3-s + (0.962 − 0.138i)5-s + (0.368 − 0.806i)7-s + (0.306 − 2.13i)9-s + (0.0234 − 0.0798i)11-s + (−1.08 + 0.493i)13-s + (−1.13 + 1.30i)15-s + (−1.16 − 0.748i)17-s + (−0.779 − 1.21i)19-s + (0.443 + 1.51i)21-s + (−0.996 − 0.0823i)23-s + (−0.0513 + 0.0150i)25-s + (1.11 + 1.72i)27-s + (0.274 − 0.426i)29-s + (0.630 − 0.727i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(736\)    =    \(2^{5} \cdot 23\)
Sign: $0.100 + 0.994i$
Analytic conductor: \(5.87698\)
Root analytic conductor: \(2.42425\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{736} (561, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 736,\ (\ :1/2),\ 0.100 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.429001 - 0.387939i\)
\(L(\frac12)\) \(\approx\) \(0.429001 - 0.387939i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 + (4.77 + 0.395i)T \)
good3 \( 1 + (2.32 - 2.01i)T + (0.426 - 2.96i)T^{2} \)
5 \( 1 + (-2.15 + 0.309i)T + (4.79 - 1.40i)T^{2} \)
7 \( 1 + (-0.974 + 2.13i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (-0.0777 + 0.264i)T + (-9.25 - 5.94i)T^{2} \)
13 \( 1 + (3.89 - 1.77i)T + (8.51 - 9.82i)T^{2} \)
17 \( 1 + (4.80 + 3.08i)T + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (3.39 + 5.28i)T + (-7.89 + 17.2i)T^{2} \)
29 \( 1 + (-1.47 + 2.29i)T + (-12.0 - 26.3i)T^{2} \)
31 \( 1 + (-3.50 + 4.04i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (0.812 + 0.116i)T + (35.5 + 10.4i)T^{2} \)
41 \( 1 + (1.14 + 7.96i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (2.46 - 2.13i)T + (6.11 - 42.5i)T^{2} \)
47 \( 1 - 10.1T + 47T^{2} \)
53 \( 1 + (-1.58 - 0.723i)T + (34.7 + 40.0i)T^{2} \)
59 \( 1 + (2.72 - 1.24i)T + (38.6 - 44.5i)T^{2} \)
61 \( 1 + (-1.80 - 1.56i)T + (8.68 + 60.3i)T^{2} \)
67 \( 1 + (4.38 + 14.9i)T + (-56.3 + 36.2i)T^{2} \)
71 \( 1 + (2.63 - 0.774i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (8.30 - 5.34i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (-3.77 - 8.27i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (8.83 + 1.27i)T + (79.6 + 23.3i)T^{2} \)
89 \( 1 + (-4.27 - 4.93i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (1.41 + 9.87i)T + (-93.0 + 27.3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.21348441436035992106407494794, −9.609667096042263212466383988590, −8.881092653510513042594363589658, −7.26338177917013427170581450810, −6.44863586636907972894227789675, −5.59736573180474120510513989617, −4.59292679860495954120260992335, −4.27304997454631845026618050135, −2.31194173063561889907362830580, −0.33264150981249411346911295048, 1.67947542632875188102574136707, 2.35428357711783955465813708454, 4.57689466575260942269152480378, 5.60348995435390260011053720669, 6.05276808598360718491660442250, 6.82228359845184101584382138290, 7.86428653557863076218945042699, 8.708229088816212854219223482856, 10.13174861773197343103371168004, 10.49884202613239583509366645348

Graph of the $Z$-function along the critical line