Properties

Label 2-76-76.7-c2-0-12
Degree $2$
Conductor $76$
Sign $0.287 + 0.957i$
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  + (−1.99 + 0.147i)2-s + (3.58 − 2.06i)3-s + (3.95 − 0.589i)4-s + (−3.72 − 6.46i)5-s + (−6.83 + 4.65i)6-s + 3.06i·7-s + (−7.80 + 1.76i)8-s + (4.05 − 7.02i)9-s + (8.39 + 12.3i)10-s − 6.31i·11-s + (12.9 − 10.2i)12-s + (8.74 − 15.1i)13-s + (−0.453 − 6.12i)14-s + (−26.7 − 15.4i)15-s + (15.3 − 4.66i)16-s + (10.6 + 18.4i)17-s + ⋯
L(s)  = 1  + (−0.997 + 0.0738i)2-s + (1.19 − 0.689i)3-s + (0.989 − 0.147i)4-s + (−0.745 − 1.29i)5-s + (−1.13 + 0.775i)6-s + 0.438i·7-s + (−0.975 + 0.220i)8-s + (0.450 − 0.780i)9-s + (0.839 + 1.23i)10-s − 0.574i·11-s + (1.07 − 0.857i)12-s + (0.672 − 1.16i)13-s + (−0.0323 − 0.437i)14-s + (−1.78 − 1.02i)15-s + (0.956 − 0.291i)16-s + (0.628 + 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.287 + 0.957i$
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: \((2,\ 76,\ (\ :1),\ 0.287 + 0.957i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.852558 - 0.634440i\)
\(L(\frac12)\) \(\approx\) \(0.852558 - 0.634440i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.99 - 0.147i)T \)
19 \( 1 + (-6.62 - 17.8i)T \)
good3 \( 1 + (-3.58 + 2.06i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (3.72 + 6.46i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 - 3.06iT - 49T^{2} \)
11 \( 1 + 6.31iT - 121T^{2} \)
13 \( 1 + (-8.74 + 15.1i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (-10.6 - 18.4i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (5.19 + 3.00i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-18.2 + 31.6i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 - 53.3iT - 961T^{2} \)
37 \( 1 + 39.2T + 1.36e3T^{2} \)
41 \( 1 + (-18.2 - 31.6i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-31.7 + 18.3i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-0.0577 - 0.0333i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (31.6 - 54.8i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (41.8 - 24.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-29.8 + 51.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (58.4 + 33.7i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-30.0 + 17.3i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (17.7 + 30.7i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (65.0 - 37.5i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 53.5iT - 6.88e3T^{2} \)
89 \( 1 + (-36.5 + 63.2i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (59.7 + 103. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−14.10106406951366262148435257866, −12.71047691836302225911256191082, −12.11704458999677574442215486148, −10.47604662453409978994932284786, −8.924118320909936412498470419620, −8.296314108219183891316194423393, −7.78794502719350460028548568627, −5.83311412326295505724706294414, −3.30401179548649989048017920598, −1.27908858533529493031370691663, 2.70278254335118704961968283706, 3.87710968814784482403090277235, 6.87368541521122599182884487900, 7.62502252467865958783904780422, 8.959556008695243657624623461047, 9.839429240872928874523195363381, 10.91669019070719649853003546741, 11.80881931269939073791766934321, 13.88642292774167124843291717994, 14.66954901600442790418829401921

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