Properties

Label 2-76-76.27-c5-0-19
Degree $2$
Conductor $76$
Sign $0.897 - 0.440i$
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  + (−5.16 − 2.30i)2-s + (−14.1 + 24.5i)3-s + (21.3 + 23.8i)4-s + (24.5 − 42.4i)5-s + (129. − 94.2i)6-s − 6.61i·7-s + (−55.4 − 172. i)8-s + (−281. − 487. i)9-s + (−224. + 162. i)10-s + 102. i·11-s + (−888. + 186. i)12-s + (718. − 415. i)13-s + (−15.2 + 34.1i)14-s + (695. + 1.20e3i)15-s + (−111. + 1.01e3i)16-s + (−44.3 + 76.8i)17-s + ⋯
L(s)  = 1  + (−0.913 − 0.407i)2-s + (−0.910 + 1.57i)3-s + (0.667 + 0.744i)4-s + (0.438 − 0.759i)5-s + (1.47 − 1.06i)6-s − 0.0510i·7-s + (−0.306 − 0.952i)8-s + (−1.15 − 2.00i)9-s + (−0.709 + 0.514i)10-s + 0.255i·11-s + (−1.78 + 0.374i)12-s + (1.17 − 0.681i)13-s + (−0.0208 + 0.0466i)14-s + (0.798 + 1.38i)15-s + (−0.108 + 0.994i)16-s + (−0.0372 + 0.0644i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 - 0.440i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.897 - 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.897 - 0.440i$
Analytic conductor: \(12.1891\)
Root analytic conductor: \(3.49129\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :5/2),\ 0.897 - 0.440i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.863465 + 0.200427i\)
\(L(\frac12)\) \(\approx\) \(0.863465 + 0.200427i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.16 + 2.30i)T \)
19 \( 1 + (-1.22e3 + 982. i)T \)
good3 \( 1 + (14.1 - 24.5i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (-24.5 + 42.4i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + 6.61iT - 1.68e4T^{2} \)
11 \( 1 - 102. iT - 1.61e5T^{2} \)
13 \( 1 + (-718. + 415. i)T + (1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (44.3 - 76.8i)T + (-7.09e5 - 1.22e6i)T^{2} \)
23 \( 1 + (1.77e3 - 1.02e3i)T + (3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (1.34e3 - 776. i)T + (1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 - 9.08e3T + 2.86e7T^{2} \)
37 \( 1 - 1.33e4iT - 6.93e7T^{2} \)
41 \( 1 + (-4.42e3 - 2.55e3i)T + (5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (6.15e3 + 3.55e3i)T + (7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (1.25e4 - 7.26e3i)T + (1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-2.02e4 + 1.17e4i)T + (2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-8.82e3 + 1.52e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (2.12e4 + 3.67e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-2.61e4 - 4.52e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (-3.59e4 + 6.21e4i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 + (-2.86e4 + 4.96e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (2.91e4 - 5.05e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 5.78e4iT - 3.93e9T^{2} \)
89 \( 1 + (-9.22e4 + 5.32e4i)T + (2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + (-1.00e5 - 5.78e4i)T + (4.29e9 + 7.43e9i)T^{2} \)
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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

−13.36809281963217975433286092435, −11.99396426680146686987114732369, −11.13981288638786046264565340455, −10.13890940326295734532878497275, −9.439815192752909087031516103051, −8.368393234495497154379711141892, −6.25410503994133515293855709018, −4.95534745956665735533955357011, −3.47100240948367996596252796756, −0.871083163506821931051729837744, 0.903644214546920513647108180773, 2.22247219960454171232820925324, 5.81113553743764029961119794959, 6.41298679113084093862175391834, 7.40406867843638741286081397556, 8.543161091200734186415630037703, 10.26678989083416744483498396913, 11.25273788305715387430058720836, 12.05494214369396631198415203755, 13.60177505093422115876798217846

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