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Properties

Label 2-368-16.5-c1-0-16
Degree $2$
Conductor $368$
Sign $0.199 - 0.979i$
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  + (0.0678 + 1.41i)2-s + (1.19 − 1.19i)3-s + (−1.99 + 0.191i)4-s + (0.672 + 0.672i)5-s + (1.77 + 1.60i)6-s + 1.79i·7-s + (−0.405 − 2.79i)8-s + 0.135i·9-s + (−0.903 + 0.994i)10-s + (2.49 + 2.49i)11-s + (−2.15 + 2.61i)12-s + (−0.0311 + 0.0311i)13-s + (−2.54 + 0.122i)14-s + 1.60·15-s + (3.92 − 0.762i)16-s + 4.24·17-s + ⋯
L(s)  = 1  + (0.0479 + 0.998i)2-s + (0.690 − 0.690i)3-s + (−0.995 + 0.0958i)4-s + (0.300 + 0.300i)5-s + (0.723 + 0.657i)6-s + 0.680i·7-s + (−0.143 − 0.989i)8-s + 0.0452i·9-s + (−0.285 + 0.314i)10-s + (0.753 + 0.753i)11-s + (−0.621 + 0.753i)12-s + (−0.00862 + 0.00862i)13-s + (−0.679 + 0.0326i)14-s + 0.415·15-s + (0.981 − 0.190i)16-s + 1.02·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.199 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.199 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $0.199 - 0.979i$
Analytic conductor: \(2.93849\)
Root analytic conductor: \(1.71420\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{368} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 368,\ (\ :1/2),\ 0.199 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28382 + 1.04885i\)
\(L(\frac12)\) \(\approx\) \(1.28382 + 1.04885i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0678 - 1.41i)T \)
23 \( 1 - iT \)
good3 \( 1 + (-1.19 + 1.19i)T - 3iT^{2} \)
5 \( 1 + (-0.672 - 0.672i)T + 5iT^{2} \)
7 \( 1 - 1.79iT - 7T^{2} \)
11 \( 1 + (-2.49 - 2.49i)T + 11iT^{2} \)
13 \( 1 + (0.0311 - 0.0311i)T - 13iT^{2} \)
17 \( 1 - 4.24T + 17T^{2} \)
19 \( 1 + (0.864 - 0.864i)T - 19iT^{2} \)
29 \( 1 + (1.05 - 1.05i)T - 29iT^{2} \)
31 \( 1 + 2.27T + 31T^{2} \)
37 \( 1 + (1.42 + 1.42i)T + 37iT^{2} \)
41 \( 1 + 8.94iT - 41T^{2} \)
43 \( 1 + (2.22 + 2.22i)T + 43iT^{2} \)
47 \( 1 + 9.40T + 47T^{2} \)
53 \( 1 + (4.89 + 4.89i)T + 53iT^{2} \)
59 \( 1 + (6.23 + 6.23i)T + 59iT^{2} \)
61 \( 1 + (-0.841 + 0.841i)T - 61iT^{2} \)
67 \( 1 + (6.47 - 6.47i)T - 67iT^{2} \)
71 \( 1 + 6.62iT - 71T^{2} \)
73 \( 1 - 0.712iT - 73T^{2} \)
79 \( 1 - 3.84T + 79T^{2} \)
83 \( 1 + (-7.62 + 7.62i)T - 83iT^{2} \)
89 \( 1 + 5.80iT - 89T^{2} \)
97 \( 1 - 19.5T + 97T^{2} \)
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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

−12.02230366521081108296006123371, −10.39267735417888742822590002843, −9.438106423335808905963154272691, −8.637664998458569544234143550505, −7.75577752214399301138333055050, −6.97731692383333059481503328111, −6.03131221159274706970645825924, −4.93732226625301715783472475202, −3.45007524292191615862839947186, −1.89793299546158056440045377543, 1.24237843368814889930926872295, 3.09173358998269879976215077889, 3.81736409660597829717095700935, 4.87306541795887202561496966457, 6.19164318771774917535962864101, 7.84364334170134135350181184829, 8.839656851146901821863345447091, 9.476110109082315564893212242121, 10.18803114990468966862578101662, 11.11990163907156563569599653494

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