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Properties

Label 2-1520-1520.949-c0-0-6
Degree $2$
Conductor $1520$
Sign $-0.382 - 0.923i$
Analytic cond. $0.758578$
Root an. cond. $0.870964$
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  + (0.707 + 0.707i)2-s + (−0.707 + 0.707i)3-s + 1.00i·4-s + 5-s − 1.00·6-s + 7-s + (−0.707 + 0.707i)8-s + (0.707 + 0.707i)10-s + (−0.707 − 0.707i)12-s + (0.707 − 0.707i)13-s + (0.707 + 0.707i)14-s + (−0.707 + 0.707i)15-s − 1.00·16-s − i·17-s + (0.707 + 0.707i)19-s + 1.00i·20-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (−0.707 + 0.707i)3-s + 1.00i·4-s + 5-s − 1.00·6-s + 7-s + (−0.707 + 0.707i)8-s + (0.707 + 0.707i)10-s + (−0.707 − 0.707i)12-s + (0.707 − 0.707i)13-s + (0.707 + 0.707i)14-s + (−0.707 + 0.707i)15-s − 1.00·16-s − i·17-s + (0.707 + 0.707i)19-s + 1.00i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $-0.382 - 0.923i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :0),\ -0.382 - 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.629751119\)
\(L(\frac12)\) \(\approx\) \(1.629751119\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 - T \)
19 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
17 \( 1 + iT - T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (1 - i)T - iT^{2} \)
47 \( 1 + 2iT - T^{2} \)
53 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
59 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
61 \( 1 + (1 - i)T - iT^{2} \)
67 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + 1.41iT - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - 1.41T + T^{2} \)
97 \( 1 - 1.41T + T^{2} \)
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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

−10.01659344284864906299583620921, −9.082364351802914821381640678919, −8.112757790347839299267319315348, −7.51556301034064104283162640601, −6.26920845530045554512115962754, −5.65747654214261524063493764849, −5.12361399422038220231657880752, −4.39548601077648099835564283187, −3.23869298987842682477544675189, −1.90630240734155895499785058029, 1.42931903073916515198860688331, 1.83351846665103561937010239225, 3.31630189675841223273804443946, 4.49140390892318177372805554304, 5.31882838924715334377866041940, 6.10290569750227607967336488275, 6.52921904796588061401896058954, 7.64263018452847633248553192393, 8.883250452811543007066618075520, 9.486367679201578794171880299377

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