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Properties

Label 2-368-16.5-c1-0-11
Degree $2$
Conductor $368$
Sign $-0.968 - 0.249i$
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  + (0.586 + 1.28i)2-s + (−0.955 + 0.955i)3-s + (−1.31 + 1.51i)4-s + (2.38 + 2.38i)5-s + (−1.79 − 0.668i)6-s − 0.198i·7-s + (−2.71 − 0.801i)8-s + 1.17i·9-s + (−1.66 + 4.46i)10-s + (−0.110 − 0.110i)11-s + (−0.189 − 2.69i)12-s + (1.69 − 1.69i)13-s + (0.255 − 0.116i)14-s − 4.55·15-s + (−0.560 − 3.96i)16-s − 0.645·17-s + ⋯
L(s)  = 1  + (0.414 + 0.909i)2-s + (−0.551 + 0.551i)3-s + (−0.655 + 0.755i)4-s + (1.06 + 1.06i)5-s + (−0.730 − 0.273i)6-s − 0.0751i·7-s + (−0.959 − 0.283i)8-s + 0.391i·9-s + (−0.527 + 1.41i)10-s + (−0.0332 − 0.0332i)11-s + (−0.0548 − 0.778i)12-s + (0.469 − 0.469i)13-s + (0.0683 − 0.0311i)14-s − 1.17·15-s + (−0.140 − 0.990i)16-s − 0.156·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.249i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $-0.968 - 0.249i$
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: \((2,\ 368,\ (\ :1/2),\ -0.968 - 0.249i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.180764 + 1.42656i\)
\(L(\frac12)\) \(\approx\) \(0.180764 + 1.42656i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.586 - 1.28i)T \)
23 \( 1 - iT \)
good3 \( 1 + (0.955 - 0.955i)T - 3iT^{2} \)
5 \( 1 + (-2.38 - 2.38i)T + 5iT^{2} \)
7 \( 1 + 0.198iT - 7T^{2} \)
11 \( 1 + (0.110 + 0.110i)T + 11iT^{2} \)
13 \( 1 + (-1.69 + 1.69i)T - 13iT^{2} \)
17 \( 1 + 0.645T + 17T^{2} \)
19 \( 1 + (-0.173 + 0.173i)T - 19iT^{2} \)
29 \( 1 + (-2.98 + 2.98i)T - 29iT^{2} \)
31 \( 1 + 4.74T + 31T^{2} \)
37 \( 1 + (1.53 + 1.53i)T + 37iT^{2} \)
41 \( 1 - 9.42iT - 41T^{2} \)
43 \( 1 + (2.46 + 2.46i)T + 43iT^{2} \)
47 \( 1 - 10.2T + 47T^{2} \)
53 \( 1 + (-6.13 - 6.13i)T + 53iT^{2} \)
59 \( 1 + (-3.18 - 3.18i)T + 59iT^{2} \)
61 \( 1 + (-0.134 + 0.134i)T - 61iT^{2} \)
67 \( 1 + (-8.34 + 8.34i)T - 67iT^{2} \)
71 \( 1 - 4.76iT - 71T^{2} \)
73 \( 1 - 5.99iT - 73T^{2} \)
79 \( 1 + 0.630T + 79T^{2} \)
83 \( 1 + (-0.711 + 0.711i)T - 83iT^{2} \)
89 \( 1 + 13.8iT - 89T^{2} \)
97 \( 1 - 7.33T + 97T^{2} \)
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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

−11.74914242063312613049080738423, −10.72964279463568956841571981009, −10.11286204138324785339682327476, −9.099821422539995033835233247054, −7.86219118722677301398056688245, −6.84616403698496361766682963791, −5.93089512161384195400221652244, −5.32462426008327453783382615952, −4.01550127516265924013384224275, −2.63362742949133578597468971440, 0.981301979300453239216381280897, 2.09662749487150721203025885434, 3.87557437151970181617503925616, 5.17898518698007257239821517490, 5.81807626669412653849735642693, 6.82129959031254348166754811545, 8.686956655309904202576388988109, 9.194756439242296117925974763129, 10.17000291884966990754959481169, 11.14955077456721127353967655464

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