Properties

Label 2-1792-1792.1077-c0-0-0
Degree $2$
Conductor $1792$
Sign $0.997 + 0.0735i$
Analytic cond. $0.894324$
Root an. cond. $0.945687$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.956 + 0.290i)2-s + (0.831 − 0.555i)4-s + (0.773 − 0.634i)7-s + (−0.634 + 0.773i)8-s + (−0.0980 − 0.995i)9-s + (−0.686 + 1.45i)11-s + (−0.555 + 0.831i)14-s + (0.382 − 0.923i)16-s + (0.382 + 0.923i)18-s + (0.235 − 1.58i)22-s + (1.90 + 0.577i)23-s + (0.881 + 0.471i)25-s + (0.290 − 0.956i)28-s + (0.163 − 0.457i)29-s + (−0.0980 + 0.995i)32-s + ⋯
L(s)  = 1  + (−0.956 + 0.290i)2-s + (0.831 − 0.555i)4-s + (0.773 − 0.634i)7-s + (−0.634 + 0.773i)8-s + (−0.0980 − 0.995i)9-s + (−0.686 + 1.45i)11-s + (−0.555 + 0.831i)14-s + (0.382 − 0.923i)16-s + (0.382 + 0.923i)18-s + (0.235 − 1.58i)22-s + (1.90 + 0.577i)23-s + (0.881 + 0.471i)25-s + (0.290 − 0.956i)28-s + (0.163 − 0.457i)29-s + (−0.0980 + 0.995i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.997 + 0.0735i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1077, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :0),\ 0.997 + 0.0735i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8063927453\)
\(L(\frac12)\) \(\approx\) \(0.8063927453\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.956 - 0.290i)T \)
7 \( 1 + (-0.773 + 0.634i)T \)
good3 \( 1 + (0.0980 + 0.995i)T^{2} \)
5 \( 1 + (-0.881 - 0.471i)T^{2} \)
11 \( 1 + (0.686 - 1.45i)T + (-0.634 - 0.773i)T^{2} \)
13 \( 1 + (0.471 + 0.881i)T^{2} \)
17 \( 1 + (-0.382 - 0.923i)T^{2} \)
19 \( 1 + (-0.290 + 0.956i)T^{2} \)
23 \( 1 + (-1.90 - 0.577i)T + (0.831 + 0.555i)T^{2} \)
29 \( 1 + (-0.163 + 0.457i)T + (-0.773 - 0.634i)T^{2} \)
31 \( 1 + (0.707 + 0.707i)T^{2} \)
37 \( 1 + (0.845 - 0.125i)T + (0.956 - 0.290i)T^{2} \)
41 \( 1 + (-0.555 + 0.831i)T^{2} \)
43 \( 1 + (-1.39 + 1.26i)T + (0.0980 - 0.995i)T^{2} \)
47 \( 1 + (0.923 - 0.382i)T^{2} \)
53 \( 1 + (0.672 + 1.88i)T + (-0.773 + 0.634i)T^{2} \)
59 \( 1 + (0.471 - 0.881i)T^{2} \)
61 \( 1 + (0.995 - 0.0980i)T^{2} \)
67 \( 1 + (1.48 - 0.0727i)T + (0.995 - 0.0980i)T^{2} \)
71 \( 1 + (-1.95 - 0.192i)T + (0.980 + 0.195i)T^{2} \)
73 \( 1 + (-0.195 - 0.980i)T^{2} \)
79 \( 1 + (-0.924 - 0.183i)T + (0.923 + 0.382i)T^{2} \)
83 \( 1 + (-0.956 - 0.290i)T^{2} \)
89 \( 1 + (-0.831 + 0.555i)T^{2} \)
97 \( 1 + (-0.707 - 0.707i)T^{2} \)
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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

−9.372299380897114005687798131118, −8.769177174462465727201039835299, −7.81402974451530744659592373013, −7.13546071621693032875474359851, −6.72706791073424947353800966315, −5.39396382813942990215183797326, −4.77366196954322043254807439395, −3.43614470865800159324329957770, −2.19872492522428193289771499014, −1.04776613942251801713021669058, 1.18144945876254005174554276565, 2.54793890182723097728753481694, 3.05439944395507447023129226467, 4.67044548672802391192920252466, 5.47087083613938866785285563056, 6.38322265585741145570799425036, 7.45961088591651253192418511588, 8.082941761852062403158701395624, 8.731564895829416313513859367715, 9.177959753267410366016558397568

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