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Properties

Label 2-76-76.75-c3-0-8
Degree $2$
Conductor $76$
Sign $-0.285 - 0.958i$
Analytic cond. $4.48414$
Root an. cond. $2.11758$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.60 + 2.33i)2-s − 3.54·3-s + (−2.87 − 7.46i)4-s + 18.5·5-s + (5.67 − 8.26i)6-s + 16.5i·7-s + (22.0 + 5.26i)8-s − 14.4·9-s + (−29.7 + 43.2i)10-s + 9.00i·11-s + (10.1 + 26.4i)12-s + 60.5i·13-s + (−38.6 − 26.5i)14-s − 65.8·15-s + (−47.5 + 42.8i)16-s + 1.33·17-s + ⋯
L(s)  = 1  + (−0.566 + 0.824i)2-s − 0.682·3-s + (−0.358 − 0.933i)4-s + 1.66·5-s + (0.386 − 0.562i)6-s + 0.894i·7-s + (0.972 + 0.232i)8-s − 0.534·9-s + (−0.940 + 1.36i)10-s + 0.246i·11-s + (0.244 + 0.636i)12-s + 1.29i·13-s + (−0.737 − 0.506i)14-s − 1.13·15-s + (−0.742 + 0.669i)16-s + 0.0189·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.285 - 0.958i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.285 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.285 - 0.958i$
Analytic conductor: \(4.48414\)
Root analytic conductor: \(2.11758\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :3/2),\ -0.285 - 0.958i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.606861 + 0.814047i\)
\(L(\frac12)\) \(\approx\) \(0.606861 + 0.814047i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.60 - 2.33i)T \)
19 \( 1 + (-82.5 - 6.40i)T \)
good3 \( 1 + 3.54T + 27T^{2} \)
5 \( 1 - 18.5T + 125T^{2} \)
7 \( 1 - 16.5iT - 343T^{2} \)
11 \( 1 - 9.00iT - 1.33e3T^{2} \)
13 \( 1 - 60.5iT - 2.19e3T^{2} \)
17 \( 1 - 1.33T + 4.91e3T^{2} \)
23 \( 1 - 182. iT - 1.21e4T^{2} \)
29 \( 1 - 61.1iT - 2.43e4T^{2} \)
31 \( 1 + 111.T + 2.97e4T^{2} \)
37 \( 1 + 160. iT - 5.06e4T^{2} \)
41 \( 1 + 386. iT - 6.89e4T^{2} \)
43 \( 1 + 300. iT - 7.95e4T^{2} \)
47 \( 1 + 206. iT - 1.03e5T^{2} \)
53 \( 1 - 544. iT - 1.48e5T^{2} \)
59 \( 1 + 444.T + 2.05e5T^{2} \)
61 \( 1 + 252.T + 2.26e5T^{2} \)
67 \( 1 - 530.T + 3.00e5T^{2} \)
71 \( 1 - 354.T + 3.57e5T^{2} \)
73 \( 1 - 343.T + 3.89e5T^{2} \)
79 \( 1 - 26.6T + 4.93e5T^{2} \)
83 \( 1 + 1.38e3iT - 5.71e5T^{2} \)
89 \( 1 - 977. iT - 7.04e5T^{2} \)
97 \( 1 + 1.18e3iT - 9.12e5T^{2} \)
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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

−14.23560723429352000056787714349, −13.69572084789213551896333989984, −12.06008351115218435608823153415, −10.79253087316920839614096132523, −9.470935106406299529919079449478, −9.005377000353346868679860140824, −7.04435878211339473174654969784, −5.79969746629150623487512831847, −5.35742602562814215772188325753, −1.85305418575520771973237732726, 0.898188293681031494518836891415, 2.85626140191063810382362830829, 5.08302642473470489803674875640, 6.41954251538738083283011347428, 8.133124755109819082595943340151, 9.575692301655179990421113007463, 10.38783905625234993199418441053, 11.14264488847091700717665138011, 12.60043615584933018670991574591, 13.44554766271937833803079584557

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