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Properties

Label 2-76-4.3-c4-0-34
Degree $2$
Conductor $76$
Sign $-0.999 + 0.0284i$
Analytic cond. $7.85611$
Root an. cond. $2.80287$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.78 − 2.86i)2-s − 10.7i·3-s + (−0.455 − 15.9i)4-s − 14.9·5-s + (−30.8 − 29.9i)6-s + 7.64i·7-s + (−47.1 − 43.2i)8-s − 34.6·9-s + (−41.6 + 42.9i)10-s + 1.00i·11-s + (−172. + 4.90i)12-s + 54.8·13-s + (21.9 + 21.3i)14-s + 160. i·15-s + (−255. + 14.5i)16-s + 79.2·17-s + ⋯
L(s)  = 1  + (0.696 − 0.717i)2-s − 1.19i·3-s + (−0.0284 − 0.999i)4-s − 0.598·5-s + (−0.856 − 0.832i)6-s + 0.155i·7-s + (−0.736 − 0.676i)8-s − 0.427·9-s + (−0.416 + 0.429i)10-s + 0.00829i·11-s + (−1.19 + 0.0340i)12-s + 0.324·13-s + (0.111 + 0.108i)14-s + 0.714i·15-s + (−0.998 + 0.0569i)16-s + 0.274·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0284i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0284i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.999 + 0.0284i$
Analytic conductor: \(7.85611\)
Root analytic conductor: \(2.80287\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :2),\ -0.999 + 0.0284i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0267401 - 1.87692i\)
\(L(\frac12)\) \(\approx\) \(0.0267401 - 1.87692i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.78 + 2.86i)T \)
19 \( 1 + 82.8iT \)
good3 \( 1 + 10.7iT - 81T^{2} \)
5 \( 1 + 14.9T + 625T^{2} \)
7 \( 1 - 7.64iT - 2.40e3T^{2} \)
11 \( 1 - 1.00iT - 1.46e4T^{2} \)
13 \( 1 - 54.8T + 2.85e4T^{2} \)
17 \( 1 - 79.2T + 8.35e4T^{2} \)
23 \( 1 + 681. iT - 2.79e5T^{2} \)
29 \( 1 - 781.T + 7.07e5T^{2} \)
31 \( 1 + 610. iT - 9.23e5T^{2} \)
37 \( 1 - 539.T + 1.87e6T^{2} \)
41 \( 1 - 766.T + 2.82e6T^{2} \)
43 \( 1 + 278. iT - 3.41e6T^{2} \)
47 \( 1 - 102. iT - 4.87e6T^{2} \)
53 \( 1 - 2.90e3T + 7.89e6T^{2} \)
59 \( 1 - 1.72e3iT - 1.21e7T^{2} \)
61 \( 1 - 5.48e3T + 1.38e7T^{2} \)
67 \( 1 - 7.29e3iT - 2.01e7T^{2} \)
71 \( 1 + 3.99e3iT - 2.54e7T^{2} \)
73 \( 1 + 7.68e3T + 2.83e7T^{2} \)
79 \( 1 - 4.51e3iT - 3.89e7T^{2} \)
83 \( 1 + 3.40e3iT - 4.74e7T^{2} \)
89 \( 1 + 6.37e3T + 6.27e7T^{2} \)
97 \( 1 + 1.61e4T + 8.85e7T^{2} \)
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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

−13.05591972022522664130211742865, −12.23033122867165028143762160199, −11.46853904648511980562634846350, −10.15564999307686116379107669695, −8.514103934628053718810516211985, −7.13193223472954421617933704593, −5.95132665096319322092110464780, −4.21243979950692372178412351773, −2.45771047629273577276960385022, −0.805789551508026180581046291633, 3.43540362912139923318424434522, 4.37047944391476553554469432705, 5.63188796676368917687137691970, 7.25537718866557458507608776415, 8.475411959936320141417521263842, 9.771145656737319724384718919194, 11.11388383539524264830718914308, 12.15578150755988525630817719851, 13.47770634535636919873541742189, 14.54604402078460579677681520182

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