Properties

Label 2-76-76.7-c2-0-15
Degree $2$
Conductor $76$
Sign $0.926 + 0.376i$
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  + 2·2-s + (1.23 − 0.715i)3-s + 4·4-s + (−3.73 − 6.47i)5-s + (2.47 − 1.43i)6-s + 3.76i·7-s + 8·8-s + (−3.47 + 6.02i)9-s + (−7.47 − 12.9i)10-s + 14.0i·11-s + (4.95 − 2.86i)12-s + (−3 + 5.19i)13-s + 7.53i·14-s + (−9.26 − 5.34i)15-s + 16·16-s + (−13.4 − 23.3i)17-s + ⋯
L(s)  = 1  + 2-s + (0.412 − 0.238i)3-s + 4-s + (−0.747 − 1.29i)5-s + (0.412 − 0.238i)6-s + 0.537i·7-s + 8-s + (−0.386 + 0.669i)9-s + (−0.747 − 1.29i)10-s + 1.27i·11-s + (0.412 − 0.238i)12-s + (−0.230 + 0.399i)13-s + 0.537i·14-s + (−0.617 − 0.356i)15-s + 16-s + (−0.792 − 1.37i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.376i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.926 + 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.926 + 0.376i$
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: \((2,\ 76,\ (\ :1),\ 0.926 + 0.376i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.12180 - 0.414691i\)
\(L(\frac12)\) \(\approx\) \(2.12180 - 0.414691i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2T \)
19 \( 1 + (-18.7 - 3.27i)T \)
good3 \( 1 + (-1.23 + 0.715i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (3.73 + 6.47i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 - 3.76iT - 49T^{2} \)
11 \( 1 - 14.0iT - 121T^{2} \)
13 \( 1 + (3 - 5.19i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (13.4 + 23.3i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (21.6 + 12.4i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-1.73 + 3.01i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + 39.4iT - 961T^{2} \)
37 \( 1 - 3.38T + 1.36e3T^{2} \)
41 \( 1 + (4.45 + 7.71i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (45.3 - 26.2i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-19.1 - 11.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-41.9 + 72.6i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (22.2 - 12.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (13.7 - 23.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-32.0 - 18.4i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-89.3 + 51.5i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-4.45 - 7.71i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (111. - 64.6i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 27.1iT - 6.88e3T^{2} \)
89 \( 1 + (64.2 - 111. i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (53.0 + 91.8i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−14.03885158090236371272430362314, −13.10494291172818084771401800337, −12.12707734898910135042538915332, −11.53083895679186798893272044775, −9.599055422352761635421039112881, −8.220506355573729991745796987706, −7.20492768639750518730099848790, −5.26597033623146415118511824267, −4.37251911802085148314727859887, −2.28454488513586563838736527553, 3.11555682435495311365076196549, 3.83287573508576673010953296209, 5.95838159015304368150552002255, 7.08132331610636919852417243452, 8.319770056994913734956758084039, 10.33914208074678793775420571991, 11.13474355151864456452129002761, 12.07675712679290601948273326017, 13.63079232813123628297919292901, 14.29194265867851138589051244953

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