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Properties

Label 2-76-4.3-c4-0-26
Degree $2$
Conductor $76$
Sign $-0.949 + 0.315i$
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  + (−3.24 − 2.34i)2-s − 7.00i·3-s + (5.04 + 15.1i)4-s + 9.28·5-s + (−16.3 + 22.7i)6-s − 54.2i·7-s + (19.1 − 61.0i)8-s + 31.9·9-s + (−30.1 − 21.7i)10-s − 65.2i·11-s + (106. − 35.3i)12-s − 57.2·13-s + (−126. + 175. i)14-s − 65.0i·15-s + (−205. + 153. i)16-s − 431.·17-s + ⋯
L(s)  = 1  + (−0.810 − 0.585i)2-s − 0.778i·3-s + (0.315 + 0.949i)4-s + 0.371·5-s + (−0.455 + 0.631i)6-s − 1.10i·7-s + (0.299 − 0.953i)8-s + 0.394·9-s + (−0.301 − 0.217i)10-s − 0.539i·11-s + (0.738 − 0.245i)12-s − 0.338·13-s + (−0.647 + 0.897i)14-s − 0.289i·15-s + (−0.801 + 0.598i)16-s − 1.49·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.315i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.949 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.949 + 0.315i$
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: \((2,\ 76,\ (\ :2),\ -0.949 + 0.315i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.149723 - 0.926035i\)
\(L(\frac12)\) \(\approx\) \(0.149723 - 0.926035i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.24 + 2.34i)T \)
19 \( 1 - 82.8iT \)
good3 \( 1 + 7.00iT - 81T^{2} \)
5 \( 1 - 9.28T + 625T^{2} \)
7 \( 1 + 54.2iT - 2.40e3T^{2} \)
11 \( 1 + 65.2iT - 1.46e4T^{2} \)
13 \( 1 + 57.2T + 2.85e4T^{2} \)
17 \( 1 + 431.T + 8.35e4T^{2} \)
23 \( 1 + 458. iT - 2.79e5T^{2} \)
29 \( 1 - 231.T + 7.07e5T^{2} \)
31 \( 1 - 195. iT - 9.23e5T^{2} \)
37 \( 1 + 1.29e3T + 1.87e6T^{2} \)
41 \( 1 - 2.13e3T + 2.82e6T^{2} \)
43 \( 1 + 1.22e3iT - 3.41e6T^{2} \)
47 \( 1 + 1.36e3iT - 4.87e6T^{2} \)
53 \( 1 - 5.21e3T + 7.89e6T^{2} \)
59 \( 1 + 1.37e3iT - 1.21e7T^{2} \)
61 \( 1 + 2.23e3T + 1.38e7T^{2} \)
67 \( 1 + 1.62e3iT - 2.01e7T^{2} \)
71 \( 1 + 175. iT - 2.54e7T^{2} \)
73 \( 1 + 2.13e3T + 2.83e7T^{2} \)
79 \( 1 - 1.20e4iT - 3.89e7T^{2} \)
83 \( 1 - 3.29e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.47e4T + 6.27e7T^{2} \)
97 \( 1 - 1.06e4T + 8.85e7T^{2} \)
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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

−13.20912277784373723631768930223, −12.15574088542854117599498218362, −10.88639382570577175546377269758, −10.05278762876120019975588758650, −8.687922094252393862126058229074, −7.45150931918547835744568724935, −6.58848621905720911150605483779, −4.10400780980834067458427286812, −2.13096528663464178911801423583, −0.61485910065012368066849884801, 2.11866516475300581575982482017, 4.68509123486423245817614273373, 5.93375983753768993663622318154, 7.32221313399423510103679923864, 8.902948268152799969935125871585, 9.525014035590283595941530276747, 10.56181131956372493637697962611, 11.79663371861190240268546608038, 13.33048366093382556056556432951, 14.78729852546547764433105024950

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