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Properties

Label 2-76-76.11-c2-0-0
Degree $2$
Conductor $76$
Sign $0.137 - 0.990i$
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 − 1.73i)2-s + (−1.23 − 0.715i)3-s + (−1.99 + 3.46i)4-s + (−3.73 + 6.47i)5-s + 2.86i·6-s + 3.76i·7-s + 7.99·8-s + (−3.47 − 6.02i)9-s + 14.9·10-s + 14.0i·11-s + (4.95 − 2.86i)12-s + (−3 − 5.19i)13-s + (6.52 − 3.76i)14-s + (9.26 − 5.34i)15-s + (−8 − 13.8i)16-s + (−13.4 + 23.3i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.412 − 0.238i)3-s + (−0.499 + 0.866i)4-s + (−0.747 + 1.29i)5-s + 0.476i·6-s + 0.537i·7-s + 0.999·8-s + (−0.386 − 0.669i)9-s + 1.49·10-s + 1.27i·11-s + (0.412 − 0.238i)12-s + (−0.230 − 0.399i)13-s + (0.465 − 0.268i)14-s + (0.617 − 0.356i)15-s + (−0.5 − 0.866i)16-s + (−0.792 + 1.37i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.137 - 0.990i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.137 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.137 - 0.990i$
Analytic conductor: \(2.07085\)
Root analytic conductor: \(1.43904\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1),\ 0.137 - 0.990i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.329092 + 0.286667i\)
\(L(\frac12)\) \(\approx\) \(0.329092 + 0.286667i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + 1.73i)T \)
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

−14.85086541603350450197005989719, −12.88256630110549889056246603392, −12.17690639955799940449453955038, −11.14414925373181616582061285753, −10.40151041902977604560963584289, −8.970064967434868018214504110193, −7.61904357394588394609833972769, −6.44263903231675646344323920329, −4.12200762065085217455916416857, −2.54045157984970087036309600163, 0.45502623186036427545334138825, 4.47200404833400817675958655885, 5.39290896132221353326462828956, 7.06988588267110842598170234306, 8.387625861817615471193328203327, 9.046077504966853677014751480462, 10.69418176380227183782373009320, 11.58497877265888347666413586551, 13.22903222241867195105363376793, 14.00725989901465322388409961026

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