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Properties

Label 2-2496-2496.77-c0-0-3
Degree $2$
Conductor $2496$
Sign $0.0980 + 0.995i$
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  + (0.881 − 0.471i)2-s + (0.831 − 0.555i)3-s + (0.555 − 0.831i)4-s + (0.924 − 0.183i)5-s + (0.471 − 0.881i)6-s + (0.0980 − 0.995i)8-s + (0.382 − 0.923i)9-s + (0.728 − 0.598i)10-s + (−1.10 + 1.65i)11-s − i·12-s + (−0.980 − 0.195i)13-s + (0.666 − 0.666i)15-s + (−0.382 − 0.923i)16-s + (−0.0980 − 0.995i)18-s + (0.360 − 0.871i)20-s + ⋯
L(s)  = 1  + (0.881 − 0.471i)2-s + (0.831 − 0.555i)3-s + (0.555 − 0.831i)4-s + (0.924 − 0.183i)5-s + (0.471 − 0.881i)6-s + (0.0980 − 0.995i)8-s + (0.382 − 0.923i)9-s + (0.728 − 0.598i)10-s + (−1.10 + 1.65i)11-s − i·12-s + (−0.980 − 0.195i)13-s + (0.666 − 0.666i)15-s + (−0.382 − 0.923i)16-s + (−0.0980 − 0.995i)18-s + (0.360 − 0.871i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0980 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0980 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2496\)    =    \(2^{6} \cdot 3 \cdot 13\)
Sign: $0.0980 + 0.995i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2496} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2496,\ (\ :0),\ 0.0980 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.836904591\)
\(L(\frac12)\) \(\approx\) \(2.836904591\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.881 + 0.471i)T \)
3 \( 1 + (-0.831 + 0.555i)T \)
13 \( 1 + (0.980 + 0.195i)T \)
good5 \( 1 + (-0.924 + 0.183i)T + (0.923 - 0.382i)T^{2} \)
7 \( 1 + (0.707 - 0.707i)T^{2} \)
11 \( 1 + (1.10 - 1.65i)T + (-0.382 - 0.923i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (0.923 + 0.382i)T^{2} \)
23 \( 1 + (-0.707 - 0.707i)T^{2} \)
29 \( 1 + (0.382 - 0.923i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (0.923 - 0.382i)T^{2} \)
41 \( 1 + (-1.76 - 0.732i)T + (0.707 + 0.707i)T^{2} \)
43 \( 1 + (-0.324 - 0.216i)T + (0.382 + 0.923i)T^{2} \)
47 \( 1 + (-0.897 - 0.897i)T + iT^{2} \)
53 \( 1 + (0.382 + 0.923i)T^{2} \)
59 \( 1 + (1.72 - 0.344i)T + (0.923 - 0.382i)T^{2} \)
61 \( 1 + (-1.53 + 1.02i)T + (0.382 - 0.923i)T^{2} \)
67 \( 1 + (-0.382 + 0.923i)T^{2} \)
71 \( 1 + (-0.0750 - 0.181i)T + (-0.707 + 0.707i)T^{2} \)
73 \( 1 + (0.707 + 0.707i)T^{2} \)
79 \( 1 + (-0.275 + 0.275i)T - iT^{2} \)
83 \( 1 + (0.113 - 0.569i)T + (-0.923 - 0.382i)T^{2} \)
89 \( 1 + (1.17 - 0.485i)T + (0.707 - 0.707i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.366213229180268347719521417692, −7.85321874528783389545606106028, −7.43361964887510821351273127332, −6.57100063116513807430328968996, −5.70631297703911253052112799292, −4.87984719461400887644674178058, −4.17648639831692945317034479657, −2.76131826528097999197817943392, −2.39028433000316999258663863432, −1.49444692482539944944917978613, 2.20676227434704902699922975601, 2.73647488053278157276499645744, 3.59787969590759856844601464068, 4.57761379504892394587953807085, 5.49451053431250100948087279510, 5.84584764227666080370544944247, 7.00653815216855521289194732844, 7.75691668979008978969870931544, 8.424153294801156431856624951688, 9.147521416984292032895360591947

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