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Properties

Label 2-76-76.7-c2-0-3
Degree $2$
Conductor $76$
Sign $-0.252 - 0.967i$
Analytic cond. $2.07085$
Root an. cond. $1.43904$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.82 + 0.813i)2-s + (−0.851 + 0.491i)3-s + (2.67 − 2.97i)4-s + (1.71 + 2.97i)5-s + (1.15 − 1.59i)6-s + 2.43i·7-s + (−2.47 + 7.60i)8-s + (−4.01 + 6.95i)9-s + (−5.55 − 4.03i)10-s + 7.76i·11-s + (−0.819 + 3.84i)12-s + (−4.63 + 8.02i)13-s + (−1.98 − 4.45i)14-s + (−2.92 − 1.68i)15-s + (−1.65 − 15.9i)16-s + (8.87 + 15.3i)17-s + ⋯
L(s)  = 1  + (−0.913 + 0.406i)2-s + (−0.283 + 0.163i)3-s + (0.669 − 0.742i)4-s + (0.343 + 0.594i)5-s + (0.192 − 0.265i)6-s + 0.348i·7-s + (−0.309 + 0.950i)8-s + (−0.446 + 0.772i)9-s + (−0.555 − 0.403i)10-s + 0.705i·11-s + (−0.0682 + 0.320i)12-s + (−0.356 + 0.617i)13-s + (−0.141 − 0.318i)14-s + (−0.194 − 0.112i)15-s + (−0.103 − 0.994i)16-s + (0.521 + 0.903i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.252 - 0.967i$
Analytic conductor: \(2.07085\)
Root analytic conductor: \(1.43904\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1),\ -0.252 - 0.967i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.443066 + 0.573798i\)
\(L(\frac12)\) \(\approx\) \(0.443066 + 0.573798i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.82 - 0.813i)T \)
19 \( 1 + (-6.84 + 17.7i)T \)
good3 \( 1 + (0.851 - 0.491i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-1.71 - 2.97i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 - 2.43iT - 49T^{2} \)
11 \( 1 - 7.76iT - 121T^{2} \)
13 \( 1 + (4.63 - 8.02i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (-8.87 - 15.3i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (-10.0 - 5.80i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-2.64 + 4.57i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + 10.2iT - 961T^{2} \)
37 \( 1 + 7.71T + 1.36e3T^{2} \)
41 \( 1 + (25.2 + 43.6i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (43.2 - 24.9i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-74.9 - 43.2i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-17.3 + 30.0i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (63.9 - 36.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-40.4 + 70.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (14.8 + 8.60i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-47.9 + 27.6i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-27.9 - 48.4i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-95.2 + 54.9i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 31.3iT - 6.88e3T^{2} \)
89 \( 1 + (5.29 - 9.17i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-65.4 - 113. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−14.80022411023744713529388678223, −13.81910869723554128765050120654, −12.07860716238944429153833376475, −10.95733439325103938463504433870, −10.11042527186006791150829444985, −8.968176478705099918253162999834, −7.62081515492945318404607805929, −6.44817487671988377595283741410, −5.14402459312669807650555285674, −2.31397859177572128822767624451, 0.886182368673908640143426385684, 3.26257676113740747121245133911, 5.57108987740095349188803499864, 7.06938297098523276287044450720, 8.413097195590370032030712098397, 9.423596435943806649051752047843, 10.51627593792092198429641920302, 11.74789255175919654941053993035, 12.51138063869937887739048971803, 13.75374902801219529481851771376

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