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Properties

Label 2-2496-24.11-c1-0-38
Degree $2$
Conductor $2496$
Sign $0.169 - 0.985i$
Analytic cond. $19.9306$
Root an. cond. $4.46437$
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  + (1 + 1.41i)3-s + 1.41·5-s − 4i·7-s + (−1.00 + 2.82i)9-s + 4.24i·11-s + i·13-s + (1.41 + 2.00i)15-s + 5.65i·17-s + 8·19-s + (5.65 − 4i)21-s − 2.82·23-s − 2.99·25-s + (−5.00 + 1.41i)27-s − 2.82·29-s + 4i·31-s + ⋯
L(s)  = 1  + (0.577 + 0.816i)3-s + 0.632·5-s − 1.51i·7-s + (−0.333 + 0.942i)9-s + 1.27i·11-s + 0.277i·13-s + (0.365 + 0.516i)15-s + 1.37i·17-s + 1.83·19-s + (1.23 − 0.872i)21-s − 0.589·23-s − 0.599·25-s + (−0.962 + 0.272i)27-s − 0.525·29-s + 0.718i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2496\)    =    \(2^{6} \cdot 3 \cdot 13\)
Sign: $0.169 - 0.985i$
Analytic conductor: \(19.9306\)
Root analytic conductor: \(4.46437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2496} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2496,\ (\ :1/2),\ 0.169 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.399140872\)
\(L(\frac12)\) \(\approx\) \(2.399140872\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1 - 1.41i)T \)
13 \( 1 - iT \)
good5 \( 1 - 1.41T + 5T^{2} \)
7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 - 4.24iT - 11T^{2} \)
17 \( 1 - 5.65iT - 17T^{2} \)
19 \( 1 - 8T + 19T^{2} \)
23 \( 1 + 2.82T + 23T^{2} \)
29 \( 1 + 2.82T + 29T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 9.89iT - 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 9.89T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 12.7iT - 59T^{2} \)
61 \( 1 + 2iT - 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 9.89T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 - 16iT - 79T^{2} \)
83 \( 1 - 1.41iT - 83T^{2} \)
89 \( 1 - 7.07iT - 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−9.410392440255489178523258066215, −8.333438017625968385952147554197, −7.48240232356431600050665380232, −7.08714635579735862011514154173, −5.83487495522077966009871262727, −5.05662949813539081983619876636, −4.02212498500552615122859575765, −3.76513983613449528922031738438, −2.36151349464036838548308437711, −1.39695518531340948189445696272, 0.76344746478560799260305192771, 2.07563112965340138932112828487, 2.79953552065868649188493781477, 3.47828600721632211842216474520, 5.15259612264485136437094433522, 5.80760643992266335725792802628, 6.20447235292407760655998781876, 7.41402903277252488627084364980, 7.964174935841737605284962074818, 8.815381227463886481785652753929

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