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Properties

Label 2-792-792.259-c1-0-25
Degree $2$
Conductor $792$
Sign $0.894 + 0.446i$
Analytic cond. $6.32415$
Root an. cond. $2.51478$
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  + (0.294 − 1.38i)2-s + (−1.51 − 0.846i)3-s + (−1.82 − 0.813i)4-s + (−1.61 + 1.84i)6-s + (−1.66 + 2.28i)8-s + (1.56 + 2.55i)9-s + (1.59 + 2.90i)11-s + (2.07 + 2.77i)12-s + (2.67 + 2.97i)16-s + (2.59 + 0.842i)17-s + (3.99 − 1.41i)18-s + (−5.03 + 6.92i)19-s + (4.49 − 1.35i)22-s + (4.44 − 2.04i)24-s + (4.56 − 2.03i)25-s + ⋯
L(s)  = 1  + (0.207 − 0.978i)2-s + (−0.872 − 0.488i)3-s + (−0.913 − 0.406i)4-s + (−0.659 + 0.751i)6-s + (−0.587 + 0.809i)8-s + (0.522 + 0.852i)9-s + (0.481 + 0.876i)11-s + (0.598 + 0.801i)12-s + (0.669 + 0.743i)16-s + (0.629 + 0.204i)17-s + (0.942 − 0.333i)18-s + (−1.15 + 1.58i)19-s + (0.957 − 0.288i)22-s + (0.908 − 0.418i)24-s + (0.913 − 0.406i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.446i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(792\)    =    \(2^{3} \cdot 3^{2} \cdot 11\)
Sign: $0.894 + 0.446i$
Analytic conductor: \(6.32415\)
Root analytic conductor: \(2.51478\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{792} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 792,\ (\ :1/2),\ 0.894 + 0.446i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.967583 - 0.227911i\)
\(L(\frac12)\) \(\approx\) \(0.967583 - 0.227911i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.294 + 1.38i)T \)
3 \( 1 + (1.51 + 0.846i)T \)
11 \( 1 + (-1.59 - 2.90i)T \)
good5 \( 1 + (-4.56 + 2.03i)T^{2} \)
7 \( 1 + (-6.84 + 1.45i)T^{2} \)
13 \( 1 + (-1.35 - 12.9i)T^{2} \)
17 \( 1 + (-2.59 - 0.842i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (5.03 - 6.92i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-28.3 + 6.02i)T^{2} \)
31 \( 1 + (3.24 + 30.8i)T^{2} \)
37 \( 1 + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (3.09 + 0.325i)T + (40.1 + 8.52i)T^{2} \)
43 \( 1 + (-8.98 - 5.18i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-31.4 + 34.9i)T^{2} \)
53 \( 1 + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (12.2 + 5.46i)T + (39.4 + 43.8i)T^{2} \)
61 \( 1 + (-6.37 + 60.6i)T^{2} \)
67 \( 1 + (-4.95 - 8.57i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-7.54 - 10.3i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (72.1 + 32.1i)T^{2} \)
83 \( 1 + (13.3 - 12.0i)T + (8.67 - 82.5i)T^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 + (-13.0 - 2.76i)T + (88.6 + 39.4i)T^{2} \)
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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

−10.36921958071417289478453997558, −9.771268721538364236967713383023, −8.597863279035984225852409754854, −7.68274754616164546854670788730, −6.50520954490784291449517624134, −5.72169690828099806880124172426, −4.69055917653327722619382868385, −3.86144113747628922982687670248, −2.27366025247448692662148037231, −1.22500381000604919850877924644, 0.61889370057050983590046637261, 3.20821155914290129899324219667, 4.28985120222548279035577809267, 5.08124130661032679877450172819, 5.99401934527265835167709419348, 6.66147569082017269350984097134, 7.52580785476144925851228516540, 8.870936567331462669795364276617, 9.145392591511108735099571370706, 10.39857682840321236057332732551

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