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Properties

Label 2-2496-104.77-c1-0-33
Degree $2$
Conductor $2496$
Sign $0.846 - 0.532i$
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  + i·3-s + 3.09·5-s + 0.646i·7-s − 9-s − 1.41·11-s + (2.44 − 2.64i)13-s + 3.09i·15-s + 3.58·17-s − 1.11·19-s − 0.646·21-s + 3.46·23-s + 4.58·25-s − i·27-s + 0.913i·29-s − 1.93i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.38·5-s + 0.244i·7-s − 0.333·9-s − 0.426·11-s + (0.679 − 0.733i)13-s + 0.799i·15-s + 0.868·17-s − 0.256·19-s − 0.140·21-s + 0.722·23-s + 0.916·25-s − 0.192i·27-s + 0.169i·29-s − 0.348i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2496\)    =    \(2^{6} \cdot 3 \cdot 13\)
Sign: $0.846 - 0.532i$
Analytic conductor: \(19.9306\)
Root analytic conductor: \(4.46437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2496} (2209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2496,\ (\ :1/2),\ 0.846 - 0.532i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.479365897\)
\(L(\frac12)\) \(\approx\) \(2.479365897\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
13 \( 1 + (-2.44 + 2.64i)T \)
good5 \( 1 - 3.09T + 5T^{2} \)
7 \( 1 - 0.646iT - 7T^{2} \)
11 \( 1 + 1.41T + 11T^{2} \)
17 \( 1 - 3.58T + 17T^{2} \)
19 \( 1 + 1.11T + 19T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 - 0.913iT - 29T^{2} \)
31 \( 1 + 1.93iT - 31T^{2} \)
37 \( 1 - 4.89T + 37T^{2} \)
41 \( 1 + 5.95iT - 41T^{2} \)
43 \( 1 - 3.58iT - 43T^{2} \)
47 \( 1 - 3.74iT - 47T^{2} \)
53 \( 1 - 10.5iT - 53T^{2} \)
59 \( 1 - 9.30T + 59T^{2} \)
61 \( 1 - 0.913iT - 61T^{2} \)
67 \( 1 - 14.6T + 67T^{2} \)
71 \( 1 + 2.44iT - 71T^{2} \)
73 \( 1 + 5.65iT - 73T^{2} \)
79 \( 1 + 14.0T + 79T^{2} \)
83 \( 1 + 11.5T + 83T^{2} \)
89 \( 1 + 8.78iT - 89T^{2} \)
97 \( 1 - 7.89iT - 97T^{2} \)
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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

−9.128314017722084838062546411275, −8.394181974168399463814464069351, −7.52991509527388453587997596428, −6.44148492268506271820850723668, −5.63305389578296727986016442341, −5.39687494703056925439083086032, −4.23181652210171847935941809916, −3.12116941602349272089942300612, −2.37299598304877726588984264119, −1.09444932672403629886654121857, 1.04223969251850735309742281238, 1.95766721924540948733858172796, 2.84171330075421180220772995909, 3.99300561923520025436859530499, 5.21152256480048859462037776548, 5.74672774416558946285325200353, 6.58464080124348871676483211450, 7.12000055767154466507510309006, 8.194375023488725779161581716222, 8.787300332921031389459124081799

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