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Properties

Label 2-2496-39.23-c0-0-1
Degree $2$
Conductor $2496$
Sign $0.0128 - 0.999i$
Analytic cond. $1.24566$
Root an. cond. $1.11609$
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)3-s + (1.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (−0.5 + 0.866i)13-s + 1.73i·21-s − 25-s − 0.999·27-s − 1.73i·31-s − 0.999·39-s + (0.5 − 0.866i)43-s + (1 + 1.73i)49-s + (0.5 − 0.866i)61-s + (−1.49 + 0.866i)63-s + (1.5 − 0.866i)67-s + 1.73i·73-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (1.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (−0.5 + 0.866i)13-s + 1.73i·21-s − 25-s − 0.999·27-s − 1.73i·31-s − 0.999·39-s + (0.5 − 0.866i)43-s + (1 + 1.73i)49-s + (0.5 − 0.866i)61-s + (−1.49 + 0.866i)63-s + (1.5 − 0.866i)67-s + 1.73i·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2496\)    =    \(2^{6} \cdot 3 \cdot 13\)
Sign: $0.0128 - 0.999i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2496} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2496,\ (\ :0),\ 0.0128 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.574340899\)
\(L(\frac12)\) \(\approx\) \(1.574340899\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + T^{2} \)
7 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + 1.73iT - T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T^{2} \)
73 \( 1 - 1.73iT - T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
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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

−9.349162135002096006809003036393, −8.487645459118216843843176447230, −8.036880504377731639970719930146, −7.23400973143624764323803907074, −5.91777541917120383364022307576, −5.26485439993429910243432686224, −4.50389817841139977668935656357, −3.84903011757822454714611796292, −2.45163591278318856529247039578, −1.93363107643259352758655575256, 1.05283801673542125128765205737, 1.96098300464191485909950230985, 3.04798926946014216688482962529, 4.07142739994042895222895715722, 4.99200654388964607915116731433, 5.80345809106591536221378854028, 6.92572137838990452273564673539, 7.47121187638193623255133366214, 8.101612447301050987904066154650, 8.570991253250151868383222416105

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