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Properties

Label 2-76-76.75-c5-0-15
Degree $2$
Conductor $76$
Sign $0.591 - 0.806i$
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  + (−5.30 − 1.95i)2-s − 7.37·3-s + (24.3 + 20.7i)4-s + 74.1·5-s + (39.1 + 14.4i)6-s + 80.5i·7-s + (−88.5 − 157. i)8-s − 188.·9-s + (−393. − 145. i)10-s − 299. i·11-s + (−179. − 153. i)12-s + 472. i·13-s + (157. − 427. i)14-s − 546.·15-s + (161. + 1.01e3i)16-s + 861.·17-s + ⋯
L(s)  = 1  + (−0.938 − 0.345i)2-s − 0.473·3-s + (0.760 + 0.648i)4-s + 1.32·5-s + (0.443 + 0.163i)6-s + 0.621i·7-s + (−0.489 − 0.872i)8-s − 0.776·9-s + (−1.24 − 0.458i)10-s − 0.747i·11-s + (−0.359 − 0.306i)12-s + 0.775i·13-s + (0.214 − 0.582i)14-s − 0.627·15-s + (0.157 + 0.987i)16-s + 0.722·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.591 - 0.806i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.591 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.591 - 0.806i$
Analytic conductor: \(12.1891\)
Root analytic conductor: \(3.49129\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :5/2),\ 0.591 - 0.806i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.911993 + 0.462067i\)
\(L(\frac12)\) \(\approx\) \(0.911993 + 0.462067i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.30 + 1.95i)T \)
19 \( 1 + (-115. - 1.56e3i)T \)
good3 \( 1 + 7.37T + 243T^{2} \)
5 \( 1 - 74.1T + 3.12e3T^{2} \)
7 \( 1 - 80.5iT - 1.68e4T^{2} \)
11 \( 1 + 299. iT - 1.61e5T^{2} \)
13 \( 1 - 472. iT - 3.71e5T^{2} \)
17 \( 1 - 861.T + 1.41e6T^{2} \)
23 \( 1 - 840. iT - 6.43e6T^{2} \)
29 \( 1 - 5.15e3iT - 2.05e7T^{2} \)
31 \( 1 - 3.42e3T + 2.86e7T^{2} \)
37 \( 1 - 4.13e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.96e4iT - 1.15e8T^{2} \)
43 \( 1 - 1.19e4iT - 1.47e8T^{2} \)
47 \( 1 + 1.58e4iT - 2.29e8T^{2} \)
53 \( 1 - 4.88e3iT - 4.18e8T^{2} \)
59 \( 1 - 3.14e4T + 7.14e8T^{2} \)
61 \( 1 + 2.94e4T + 8.44e8T^{2} \)
67 \( 1 + 2.17e4T + 1.35e9T^{2} \)
71 \( 1 - 2.65e3T + 1.80e9T^{2} \)
73 \( 1 + 4.62e4T + 2.07e9T^{2} \)
79 \( 1 - 7.59e4T + 3.07e9T^{2} \)
83 \( 1 - 3.50e4iT - 3.93e9T^{2} \)
89 \( 1 - 6.13e4iT - 5.58e9T^{2} \)
97 \( 1 + 9.09e4iT - 8.58e9T^{2} \)
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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

−13.61424161545622082719741864352, −12.24469532217716347862541720433, −11.38851251092187056450358215648, −10.22853625304751658130985324184, −9.280320314074944958974334646632, −8.274212392825912752477138277576, −6.43203749975245075656429417111, −5.60593995611278185147822527976, −2.93390890785525698275771773800, −1.43341559530557633738355811361, 0.66726982752938189616975359748, 2.40469813338601967775998288441, 5.28441553318858006821417560533, 6.22308217527472471906546035343, 7.48619694932313260944004594508, 8.948274551695323053500595736135, 10.04262500178904910658567058817, 10.70491274817853382181378427940, 12.06294027109846311870797948860, 13.55308124950234351034985839822

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