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Properties

Label 2-392-392.29-c1-0-24
Degree $2$
Conductor $392$
Sign $0.999 + 0.0251i$
Analytic cond. $3.13013$
Root an. cond. $1.76921$
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  + (−1.27 + 0.613i)2-s + (−0.572 + 0.130i)3-s + (1.24 − 1.56i)4-s + (2.54 − 0.580i)5-s + (0.649 − 0.517i)6-s + (2.64 + 0.117i)7-s + (−0.630 + 2.75i)8-s + (−2.39 + 1.15i)9-s + (−2.88 + 2.30i)10-s + (2.31 − 4.80i)11-s + (−0.509 + 1.05i)12-s + (0.178 − 0.369i)13-s + (−3.44 + 1.47i)14-s + (−1.38 + 0.665i)15-s + (−0.888 − 3.90i)16-s + (−0.776 + 0.974i)17-s + ⋯
L(s)  = 1  + (−0.901 + 0.433i)2-s + (−0.330 + 0.0754i)3-s + (0.623 − 0.781i)4-s + (1.13 − 0.259i)5-s + (0.265 − 0.211i)6-s + (0.999 + 0.0444i)7-s + (−0.222 + 0.974i)8-s + (−0.797 + 0.383i)9-s + (−0.912 + 0.727i)10-s + (0.698 − 1.44i)11-s + (−0.147 + 0.305i)12-s + (0.0494 − 0.102i)13-s + (−0.919 + 0.393i)14-s + (−0.356 + 0.171i)15-s + (−0.222 − 0.975i)16-s + (−0.188 + 0.236i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0251i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0251i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.999 + 0.0251i$
Analytic conductor: \(3.13013\)
Root analytic conductor: \(1.76921\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1/2),\ 0.999 + 0.0251i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06616 - 0.0133918i\)
\(L(\frac12)\) \(\approx\) \(1.06616 - 0.0133918i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.27 - 0.613i)T \)
7 \( 1 + (-2.64 - 0.117i)T \)
good3 \( 1 + (0.572 - 0.130i)T + (2.70 - 1.30i)T^{2} \)
5 \( 1 + (-2.54 + 0.580i)T + (4.50 - 2.16i)T^{2} \)
11 \( 1 + (-2.31 + 4.80i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (-0.178 + 0.369i)T + (-8.10 - 10.1i)T^{2} \)
17 \( 1 + (0.776 - 0.974i)T + (-3.78 - 16.5i)T^{2} \)
19 \( 1 + 2.14iT - 19T^{2} \)
23 \( 1 + (-0.363 - 0.455i)T + (-5.11 + 22.4i)T^{2} \)
29 \( 1 + (3.84 + 3.06i)T + (6.45 + 28.2i)T^{2} \)
31 \( 1 - 9.09T + 31T^{2} \)
37 \( 1 + (-3.25 - 2.59i)T + (8.23 + 36.0i)T^{2} \)
41 \( 1 + (-1.85 - 8.13i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (-12.1 - 2.76i)T + (38.7 + 18.6i)T^{2} \)
47 \( 1 + (9.15 + 4.40i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (0.0313 - 0.0250i)T + (11.7 - 51.6i)T^{2} \)
59 \( 1 + (0.932 + 0.212i)T + (53.1 + 25.5i)T^{2} \)
61 \( 1 + (5.46 + 4.35i)T + (13.5 + 59.4i)T^{2} \)
67 \( 1 - 3.98iT - 67T^{2} \)
71 \( 1 + (0.111 + 0.140i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-0.931 + 0.448i)T + (45.5 - 57.0i)T^{2} \)
79 \( 1 + 13.3T + 79T^{2} \)
83 \( 1 + (3.82 + 7.95i)T + (-51.7 + 64.8i)T^{2} \)
89 \( 1 + (11.6 - 5.60i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 - 16.9T + 97T^{2} \)
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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

−11.34652406749299715989205964326, −10.33977231447829232967321995758, −9.317363462242238154053701264714, −8.575639522298649466716766753142, −7.87682100063517596319716567939, −6.26470588747150926507769439591, −5.86888770467678788104920300189, −4.84054100471683222261248179999, −2.58575210222148020902561584046, −1.16565222191574878336064172734, 1.50294115729853270981010663765, 2.53264858075283892594102164499, 4.29511774534811221924440695134, 5.73208440603991475724633099773, 6.69589985526082859198837714310, 7.64257922539135370437544220576, 8.830085183425345372290598919858, 9.513713331596557276612639539079, 10.35334777724630682497178956878, 11.20970036984371868436209561108

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