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Properties

Label 2-1519-217.123-c0-0-3
Degree $2$
Conductor $1519$
Sign $0.968 + 0.250i$
Analytic cond. $0.758079$
Root an. cond. $0.870677$
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  + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)5-s + 8-s + (−0.5 − 0.866i)9-s + (0.499 − 0.866i)10-s + (0.5 + 0.866i)16-s + (0.499 − 0.866i)18-s + (−0.5 − 0.866i)19-s + (0.5 − 0.866i)31-s + (0.499 − 0.866i)38-s + (−0.499 − 0.866i)40-s + 41-s + (−0.499 + 0.866i)45-s + (1 + 1.73i)47-s + (−0.5 + 0.866i)59-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)5-s + 8-s + (−0.5 − 0.866i)9-s + (0.499 − 0.866i)10-s + (0.5 + 0.866i)16-s + (0.499 − 0.866i)18-s + (−0.5 − 0.866i)19-s + (0.5 − 0.866i)31-s + (0.499 − 0.866i)38-s + (−0.499 − 0.866i)40-s + 41-s + (−0.499 + 0.866i)45-s + (1 + 1.73i)47-s + (−0.5 + 0.866i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1519\)    =    \(7^{2} \cdot 31\)
Sign: $0.968 + 0.250i$
Analytic conductor: \(0.758079\)
Root analytic conductor: \(0.870677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1519} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1519,\ (\ :0),\ 0.968 + 0.250i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.379285589\)
\(L(\frac12)\) \(\approx\) \(1.379285589\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
31 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
3 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−9.356040631438886640778695076989, −8.743879257593704398437238735820, −7.906186467820005687428357321296, −7.14542225264980719802034565323, −6.22421965381850730879239178216, −5.68019894342069656689627642347, −4.57982136357682714328017470306, −4.14707869902630960258815201170, −2.68790291850424569650232601203, −1.00892909238363937758976610376, 1.84312791933419256143066304880, 2.78809519777782555222884992936, 3.51969767269653018714950428693, 4.41773479427198656877679632591, 5.37159574964039234795200812692, 6.48425072733216374184969933383, 7.42579268416548263474977017681, 7.944948598441943294811610760986, 8.860705203322919815819196350857, 10.18120124247099726860934638272

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