Properties

Label 2-792-88.43-c1-0-56
Degree $2$
Conductor $792$
Sign $-0.979 - 0.202i$
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  + (0.927 − 1.06i)2-s + (−0.280 − 1.98i)4-s − 2i·5-s − 0.936·7-s + (−2.37 − 1.53i)8-s + (−2.13 − 1.85i)10-s + (−3.09 + 1.19i)11-s − 4.27·13-s + (−0.868 + 0.999i)14-s + (−3.84 + 1.11i)16-s − 3.33i·17-s + 2.89i·19-s + (−3.96 + 0.561i)20-s + (−1.58 + 4.41i)22-s − 5.12i·23-s + ⋯
L(s)  = 1  + (0.655 − 0.755i)2-s + (−0.140 − 0.990i)4-s − 0.894i·5-s − 0.353·7-s + (−0.839 − 0.543i)8-s + (−0.675 − 0.586i)10-s + (−0.932 + 0.361i)11-s − 1.18·13-s + (−0.232 + 0.267i)14-s + (−0.960 + 0.277i)16-s − 0.808i·17-s + 0.664i·19-s + (−0.885 + 0.125i)20-s + (−0.338 + 0.941i)22-s − 1.06i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.202i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 - 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(792\)    =    \(2^{3} \cdot 3^{2} \cdot 11\)
Sign: $-0.979 - 0.202i$
Analytic conductor: \(6.32415\)
Root analytic conductor: \(2.51478\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{792} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 792,\ (\ :1/2),\ -0.979 - 0.202i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.125775 + 1.22812i\)
\(L(\frac12)\) \(\approx\) \(0.125775 + 1.22812i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.927 + 1.06i)T \)
3 \( 1 \)
11 \( 1 + (3.09 - 1.19i)T \)
good5 \( 1 + 2iT - 5T^{2} \)
7 \( 1 + 0.936T + 7T^{2} \)
13 \( 1 + 4.27T + 13T^{2} \)
17 \( 1 + 3.33iT - 17T^{2} \)
19 \( 1 - 2.89iT - 19T^{2} \)
23 \( 1 + 5.12iT - 23T^{2} \)
29 \( 1 - 6.60T + 29T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
37 \( 1 + 6.18iT - 37T^{2} \)
41 \( 1 + 1.46iT - 41T^{2} \)
43 \( 1 - 10.3iT - 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 + 8.24iT - 53T^{2} \)
59 \( 1 - 9.65T + 59T^{2} \)
61 \( 1 + 9.06T + 61T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 + 4.24iT - 71T^{2} \)
73 \( 1 + 13.2iT - 73T^{2} \)
79 \( 1 - 3.86T + 79T^{2} \)
83 \( 1 + 2.39iT - 83T^{2} \)
89 \( 1 - 3.47T + 89T^{2} \)
97 \( 1 - 11.3T + 97T^{2} \)
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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

−9.960579314116078127606995395898, −9.264713272982742262428283522475, −8.251047416618736331816970458393, −7.14378820159148612830620008794, −6.03564141384180499154768906295, −4.91421147615412089770067671177, −4.63384944886600251641453491881, −3.11593781446556235103838107406, −2.14778982151618104855725758921, −0.46550300119437706498428161548, 2.60585713885400929945633414482, 3.26123572230603238888907430760, 4.58357982854319480849363911288, 5.47459274427246635757809087977, 6.44162738053346898921590320365, 7.15703193874331400598437051624, 7.88724694585704521258103127611, 8.837408350390453684665692586056, 9.956320010210280616128099944928, 10.70959221308989809050965042764

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