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Properties

Label 2-76-76.75-c3-0-7
Degree $2$
Conductor $76$
Sign $0.900 - 0.434i$
Analytic cond. $4.48414$
Root an. cond. $2.11758$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.25 + 1.71i)2-s − 8.24·3-s + (2.15 − 7.70i)4-s − 10.8·5-s + (18.5 − 14.0i)6-s − 8.32i·7-s + (8.32 + 21.0i)8-s + 40.9·9-s + (24.4 − 18.5i)10-s + 53.9i·11-s + (−17.7 + 63.5i)12-s − 43.7i·13-s + (14.2 + 18.7i)14-s + 89.3·15-s + (−54.7 − 33.1i)16-s + 91.7·17-s + ⋯
L(s)  = 1  + (−0.796 + 0.604i)2-s − 1.58·3-s + (0.268 − 0.963i)4-s − 0.968·5-s + (1.26 − 0.959i)6-s − 0.449i·7-s + (0.368 + 0.929i)8-s + 1.51·9-s + (0.771 − 0.585i)10-s + 1.47i·11-s + (−0.426 + 1.52i)12-s − 0.933i·13-s + (0.271 + 0.358i)14-s + 1.53·15-s + (−0.855 − 0.518i)16-s + 1.30·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.434i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.900 - 0.434i$
Analytic conductor: \(4.48414\)
Root analytic conductor: \(2.11758\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :3/2),\ 0.900 - 0.434i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.429564 + 0.0983111i\)
\(L(\frac12)\) \(\approx\) \(0.429564 + 0.0983111i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.25 - 1.71i)T \)
19 \( 1 + (-54.7 - 62.1i)T \)
good3 \( 1 + 8.24T + 27T^{2} \)
5 \( 1 + 10.8T + 125T^{2} \)
7 \( 1 + 8.32iT - 343T^{2} \)
11 \( 1 - 53.9iT - 1.33e3T^{2} \)
13 \( 1 + 43.7iT - 2.19e3T^{2} \)
17 \( 1 - 91.7T + 4.91e3T^{2} \)
23 \( 1 + 172. iT - 1.21e4T^{2} \)
29 \( 1 + 3.12iT - 2.43e4T^{2} \)
31 \( 1 - 113.T + 2.97e4T^{2} \)
37 \( 1 + 159. iT - 5.06e4T^{2} \)
41 \( 1 - 205. iT - 6.89e4T^{2} \)
43 \( 1 - 28.4iT - 7.95e4T^{2} \)
47 \( 1 - 62.4iT - 1.03e5T^{2} \)
53 \( 1 - 650. iT - 1.48e5T^{2} \)
59 \( 1 - 792.T + 2.05e5T^{2} \)
61 \( 1 - 30.4T + 2.26e5T^{2} \)
67 \( 1 + 278.T + 3.00e5T^{2} \)
71 \( 1 + 74.0T + 3.57e5T^{2} \)
73 \( 1 - 1.11e3T + 3.89e5T^{2} \)
79 \( 1 - 51.9T + 4.93e5T^{2} \)
83 \( 1 + 1.13e3iT - 5.71e5T^{2} \)
89 \( 1 + 1.03e3iT - 7.04e5T^{2} \)
97 \( 1 + 145. iT - 9.12e5T^{2} \)
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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

−14.53095595178347977533588331967, −12.50792990480848056706763633000, −11.81318832174062269539830872072, −10.55618066308511443550576779454, −9.972058607151918061535668010280, −7.920195496293349212766022350713, −7.14131892102963711667501147025, −5.80596986046341625087875764418, −4.57778021803852775624091134016, −0.76250540235369401084124984509, 0.78544700900919347929026103348, 3.62033973553984886473166754049, 5.46226045051303840667996276278, 6.93468239473121430083195232212, 8.218124616418735477822029133141, 9.645877801794086418016825400772, 10.99985324215396304418417633877, 11.69851510421584967512248018191, 12.01302945200549498362177507858, 13.56687519426870366670410139745

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