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Properties

Label 2-368-23.9-c1-0-4
Degree $2$
Conductor $368$
Sign $0.959 + 0.282i$
Analytic cond. $2.93849$
Root an. cond. $1.71420$
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  + (−0.712 − 0.822i)3-s + (0.174 + 1.21i)5-s + (−0.260 + 0.570i)7-s + (0.258 − 1.79i)9-s + (5.65 + 1.65i)11-s + (−0.977 − 2.13i)13-s + (0.875 − 1.01i)15-s + (4.49 + 2.88i)17-s + (0.227 − 0.146i)19-s + (0.654 − 0.192i)21-s + (2.50 − 4.09i)23-s + (3.34 − 0.983i)25-s + (−4.40 + 2.83i)27-s + (−1.66 − 1.07i)29-s + (4.32 − 4.99i)31-s + ⋯
L(s)  = 1  + (−0.411 − 0.474i)3-s + (0.0782 + 0.543i)5-s + (−0.0984 + 0.215i)7-s + (0.0861 − 0.599i)9-s + (1.70 + 0.500i)11-s + (−0.270 − 0.593i)13-s + (0.226 − 0.260i)15-s + (1.08 + 0.700i)17-s + (0.0522 − 0.0336i)19-s + (0.142 − 0.0419i)21-s + (0.521 − 0.853i)23-s + (0.669 − 0.196i)25-s + (−0.848 + 0.545i)27-s + (−0.309 − 0.199i)29-s + (0.776 − 0.896i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.282i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 + 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $0.959 + 0.282i$
Analytic conductor: \(2.93849\)
Root analytic conductor: \(1.71420\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{368} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 368,\ (\ :1/2),\ 0.959 + 0.282i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29094 - 0.185889i\)
\(L(\frac12)\) \(\approx\) \(1.29094 - 0.185889i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.50 + 4.09i)T \)
good3 \( 1 + (0.712 + 0.822i)T + (-0.426 + 2.96i)T^{2} \)
5 \( 1 + (-0.174 - 1.21i)T + (-4.79 + 1.40i)T^{2} \)
7 \( 1 + (0.260 - 0.570i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (-5.65 - 1.65i)T + (9.25 + 5.94i)T^{2} \)
13 \( 1 + (0.977 + 2.13i)T + (-8.51 + 9.82i)T^{2} \)
17 \( 1 + (-4.49 - 2.88i)T + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (-0.227 + 0.146i)T + (7.89 - 17.2i)T^{2} \)
29 \( 1 + (1.66 + 1.07i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (-4.32 + 4.99i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (1.43 - 9.96i)T + (-35.5 - 10.4i)T^{2} \)
41 \( 1 + (1.53 + 10.7i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (-3.01 - 3.47i)T + (-6.11 + 42.5i)T^{2} \)
47 \( 1 + 6.97T + 47T^{2} \)
53 \( 1 + (1.09 - 2.39i)T + (-34.7 - 40.0i)T^{2} \)
59 \( 1 + (0.685 + 1.50i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (5.98 - 6.90i)T + (-8.68 - 60.3i)T^{2} \)
67 \( 1 + (5.94 - 1.74i)T + (56.3 - 36.2i)T^{2} \)
71 \( 1 + (8.69 - 2.55i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (4.99 - 3.21i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (-2.49 - 5.46i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (0.277 - 1.93i)T + (-79.6 - 23.3i)T^{2} \)
89 \( 1 + (3.00 + 3.46i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (1.44 + 10.0i)T + (-93.0 + 27.3i)T^{2} \)
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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

−11.59957639026940168641904849895, −10.42375711748637767921481714342, −9.616706910817128532749155334790, −8.626427521718114342472706027066, −7.34206862436637013194021893671, −6.55252325274564361544588103514, −5.85587173230390042555404601693, −4.29238279296566490149638968267, −3.05590850203942192924579865739, −1.26490196540121502458125533705, 1.35816200783005785764024973189, 3.42041762950278666091615937885, 4.58083112418508335612773906313, 5.44173338428150309921992917872, 6.62889195566904471457758530995, 7.65623640745457673955250682787, 8.965998184958868519142957635566, 9.508651600166605147461396559692, 10.56355734571462954338078777013, 11.53510845641149131638980594343

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