Properties

Label 2-76-4.3-c4-0-12
Degree $2$
Conductor $76$
Sign $0.292 - 0.956i$
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  + (−0.591 + 3.95i)2-s − 4.43i·3-s + (−15.3 − 4.67i)4-s + 13.1·5-s + (17.5 + 2.62i)6-s + 8.96i·7-s + (27.5 − 57.7i)8-s + 61.3·9-s + (−7.74 + 51.8i)10-s + 193. i·11-s + (−20.7 + 67.8i)12-s + 173.·13-s + (−35.4 − 5.30i)14-s − 58.1i·15-s + (212. + 143. i)16-s + 83.2·17-s + ⋯
L(s)  = 1  + (−0.147 + 0.989i)2-s − 0.492i·3-s + (−0.956 − 0.292i)4-s + 0.524·5-s + (0.487 + 0.0728i)6-s + 0.183i·7-s + (0.430 − 0.902i)8-s + 0.757·9-s + (−0.0774 + 0.518i)10-s + 1.60i·11-s + (−0.144 + 0.471i)12-s + 1.02·13-s + (−0.181 − 0.0270i)14-s − 0.258i·15-s + (0.828 + 0.559i)16-s + 0.288·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.292 - 0.956i$
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: \((2,\ 76,\ (\ :2),\ 0.292 - 0.956i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.29682 + 0.959558i\)
\(L(\frac12)\) \(\approx\) \(1.29682 + 0.959558i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.591 - 3.95i)T \)
19 \( 1 - 82.8iT \)
good3 \( 1 + 4.43iT - 81T^{2} \)
5 \( 1 - 13.1T + 625T^{2} \)
7 \( 1 - 8.96iT - 2.40e3T^{2} \)
11 \( 1 - 193. iT - 1.46e4T^{2} \)
13 \( 1 - 173.T + 2.85e4T^{2} \)
17 \( 1 - 83.2T + 8.35e4T^{2} \)
23 \( 1 - 684. iT - 2.79e5T^{2} \)
29 \( 1 - 1.43e3T + 7.07e5T^{2} \)
31 \( 1 + 750. iT - 9.23e5T^{2} \)
37 \( 1 - 437.T + 1.87e6T^{2} \)
41 \( 1 - 895.T + 2.82e6T^{2} \)
43 \( 1 + 928. iT - 3.41e6T^{2} \)
47 \( 1 + 2.11e3iT - 4.87e6T^{2} \)
53 \( 1 + 3.58e3T + 7.89e6T^{2} \)
59 \( 1 + 306. iT - 1.21e7T^{2} \)
61 \( 1 + 7.09e3T + 1.38e7T^{2} \)
67 \( 1 - 4.56e3iT - 2.01e7T^{2} \)
71 \( 1 + 149. iT - 2.54e7T^{2} \)
73 \( 1 + 2.82e3T + 2.83e7T^{2} \)
79 \( 1 - 6.89e3iT - 3.89e7T^{2} \)
83 \( 1 + 7.84e3iT - 4.74e7T^{2} \)
89 \( 1 + 8.71e3T + 6.27e7T^{2} \)
97 \( 1 + 2.17e3T + 8.85e7T^{2} \)
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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

−13.94100134599979158779436808976, −13.17424356716419836437807869603, −12.16195439757423528827997029917, −10.20481861534473774842825932292, −9.421115254988940967921317700524, −7.939031474868509598259479812353, −6.96231067008053316061936974961, −5.81953595600877153971760083022, −4.30275657478037660597959461766, −1.52537212064785729892270648952, 1.08061512364589354540739467730, 3.13001905177984786114481320750, 4.48811336655989142007723214343, 6.10686239702446974298796468670, 8.216064385980317278544916231083, 9.225159280277222179505447717865, 10.41488792683484374501328454035, 11.00998382690325840312140581254, 12.41357853957390639087610809925, 13.53836684010208327426760024926

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