Properties

Label 2-76-76.75-c5-0-13
Degree $2$
Conductor $76$
Sign $0.999 - 0.000726i$
Analytic cond. $12.1891$
Root an. cond. $3.49129$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.28 − 5.50i)2-s − 16.4·3-s + (−28.6 − 14.1i)4-s + 16.6·5-s + (−21.1 + 90.5i)6-s + 54.8i·7-s + (−114. + 139. i)8-s + 27.3·9-s + (21.4 − 91.8i)10-s + 545. i·11-s + (471. + 232. i)12-s − 885. i·13-s + (302. + 70.5i)14-s − 274.·15-s + (623. + 812. i)16-s + 2.30e3·17-s + ⋯
L(s)  = 1  + (0.227 − 0.973i)2-s − 1.05·3-s + (−0.896 − 0.442i)4-s + 0.298·5-s + (−0.239 + 1.02i)6-s + 0.423i·7-s + (−0.634 + 0.772i)8-s + 0.112·9-s + (0.0677 − 0.290i)10-s + 1.35i·11-s + (0.945 + 0.466i)12-s − 1.45i·13-s + (0.412 + 0.0961i)14-s − 0.314·15-s + (0.608 + 0.793i)16-s + 1.93·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.000726i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.999 - 0.000726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.999 - 0.000726i$
Analytic conductor: \(12.1891\)
Root analytic conductor: \(3.49129\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :5/2),\ 0.999 - 0.000726i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.03597 + 0.000376456i\)
\(L(\frac12)\) \(\approx\) \(1.03597 + 0.000376456i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.28 + 5.50i)T \)
19 \( 1 + (-1.41e3 + 697. i)T \)
good3 \( 1 + 16.4T + 243T^{2} \)
5 \( 1 - 16.6T + 3.12e3T^{2} \)
7 \( 1 - 54.8iT - 1.68e4T^{2} \)
11 \( 1 - 545. iT - 1.61e5T^{2} \)
13 \( 1 + 885. iT - 3.71e5T^{2} \)
17 \( 1 - 2.30e3T + 1.41e6T^{2} \)
23 \( 1 - 3.45e3iT - 6.43e6T^{2} \)
29 \( 1 - 2.57e3iT - 2.05e7T^{2} \)
31 \( 1 + 4.68e3T + 2.86e7T^{2} \)
37 \( 1 - 9.75e3iT - 6.93e7T^{2} \)
41 \( 1 - 8.78e3iT - 1.15e8T^{2} \)
43 \( 1 - 1.64e4iT - 1.47e8T^{2} \)
47 \( 1 + 7.32e3iT - 2.29e8T^{2} \)
53 \( 1 - 1.13e4iT - 4.18e8T^{2} \)
59 \( 1 + 2.59e4T + 7.14e8T^{2} \)
61 \( 1 - 4.29e4T + 8.44e8T^{2} \)
67 \( 1 + 7.12e3T + 1.35e9T^{2} \)
71 \( 1 - 1.83e4T + 1.80e9T^{2} \)
73 \( 1 + 1.36e4T + 2.07e9T^{2} \)
79 \( 1 + 4.31e4T + 3.07e9T^{2} \)
83 \( 1 + 2.69e3iT - 3.93e9T^{2} \)
89 \( 1 - 4.20e4iT - 5.58e9T^{2} \)
97 \( 1 - 3.99e4iT - 8.58e9T^{2} \)
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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

−13.10305880015312588909877314000, −12.22109066055588154129567137855, −11.54365989378665117158764957786, −10.24797748104623048641080494665, −9.592247263124441448903618081457, −7.74695979556001980792714155405, −5.70537451803341680972900553553, −5.14540538354992395535577267833, −3.13395675377505248772546386681, −1.22061341012411299898958141859, 0.57271557458300847447770164852, 3.77649626107564159555025004663, 5.43375000153244695604647928334, 6.10147648172423294227912873023, 7.40964261670723669765949995062, 8.800352975333772836561651371133, 10.14039159939186256725906793527, 11.51345987592188315029573136088, 12.41721692435239990501107350784, 14.03306283304668649838199468929

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