Properties

Label 2-1792-1792.349-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  + (−0.290 + 0.956i)2-s + (−0.831 − 0.555i)4-s + (0.634 − 0.773i)7-s + (0.773 − 0.634i)8-s + (−0.995 − 0.0980i)9-s + (0.276 + 0.0988i)11-s + (0.555 + 0.831i)14-s + (0.382 + 0.923i)16-s + (0.382 − 0.923i)18-s + (−0.174 + 0.235i)22-s + (−0.0569 − 0.187i)23-s + (−0.471 − 0.881i)25-s + (−0.956 + 0.290i)28-s + (0.733 − 1.55i)29-s + (−0.995 + 0.0980i)32-s + ⋯
L(s)  = 1  + (−0.290 + 0.956i)2-s + (−0.831 − 0.555i)4-s + (0.634 − 0.773i)7-s + (0.773 − 0.634i)8-s + (−0.995 − 0.0980i)9-s + (0.276 + 0.0988i)11-s + (0.555 + 0.831i)14-s + (0.382 + 0.923i)16-s + (0.382 − 0.923i)18-s + (−0.174 + 0.235i)22-s + (−0.0569 − 0.187i)23-s + (−0.471 − 0.881i)25-s + (−0.956 + 0.290i)28-s + (0.733 − 1.55i)29-s + (−0.995 + 0.0980i)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: $\chi_{1792} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :0),\ 0.997 - 0.0735i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8747150156\)
\(L(\frac12)\) \(\approx\) \(0.8747150156\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.290 - 0.956i)T \)
7 \( 1 + (-0.634 + 0.773i)T \)
good3 \( 1 + (0.995 + 0.0980i)T^{2} \)
5 \( 1 + (0.471 + 0.881i)T^{2} \)
11 \( 1 + (-0.276 - 0.0988i)T + (0.773 + 0.634i)T^{2} \)
13 \( 1 + (0.881 + 0.471i)T^{2} \)
17 \( 1 + (-0.382 + 0.923i)T^{2} \)
19 \( 1 + (0.956 - 0.290i)T^{2} \)
23 \( 1 + (0.0569 + 0.187i)T + (-0.831 + 0.555i)T^{2} \)
29 \( 1 + (-0.733 + 1.55i)T + (-0.634 - 0.773i)T^{2} \)
31 \( 1 + (0.707 - 0.707i)T^{2} \)
37 \( 1 + (-1.51 + 1.12i)T + (0.290 - 0.956i)T^{2} \)
41 \( 1 + (0.555 + 0.831i)T^{2} \)
43 \( 1 + (-1.80 + 0.0887i)T + (0.995 - 0.0980i)T^{2} \)
47 \( 1 + (0.923 + 0.382i)T^{2} \)
53 \( 1 + (-0.574 - 1.21i)T + (-0.634 + 0.773i)T^{2} \)
59 \( 1 + (0.881 - 0.471i)T^{2} \)
61 \( 1 + (-0.0980 + 0.995i)T^{2} \)
67 \( 1 + (1.34 + 1.48i)T + (-0.0980 + 0.995i)T^{2} \)
71 \( 1 + (-0.192 - 1.95i)T + (-0.980 + 0.195i)T^{2} \)
73 \( 1 + (0.195 - 0.980i)T^{2} \)
79 \( 1 + (1.72 - 0.344i)T + (0.923 - 0.382i)T^{2} \)
83 \( 1 + (-0.290 - 0.956i)T^{2} \)
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

−9.311160443380170182656561216616, −8.519230324853584896563268819211, −7.87353998175638458257494176943, −7.27201105546157585607847228828, −6.17854705466602228378688851500, −5.75163469872377877555235002196, −4.51965266820051717305404759343, −4.04667586122016071077396695881, −2.45588029269052089221955531227, −0.822397425135756734838628018457, 1.38525057359656916339911514715, 2.52833570476118245873215992782, 3.26608363194720382466081419230, 4.44928654779107017003644175603, 5.28411432843329923934574571726, 6.03807833230878312589520865366, 7.39001604844088974488186894212, 8.190863066331984867575693465786, 8.824603287172594680258367433196, 9.330290422731963184001988606346

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