Properties

Label 2-76-76.75-c3-0-9
Degree $2$
Conductor $76$
Sign $0.112 - 0.993i$
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  + (2.63 + 1.01i)2-s − 6.23·3-s + (5.92 + 5.37i)4-s + 8.54·5-s + (−16.4 − 6.35i)6-s + 28.2i·7-s + (10.1 + 20.2i)8-s + 11.9·9-s + (22.5 + 8.70i)10-s + 12.2i·11-s + (−36.9 − 33.5i)12-s + 3.60i·13-s + (−28.7 + 74.4i)14-s − 53.2·15-s + (6.15 + 63.7i)16-s + 100.·17-s + ⋯
L(s)  = 1  + (0.932 + 0.360i)2-s − 1.20·3-s + (0.740 + 0.672i)4-s + 0.763·5-s + (−1.11 − 0.432i)6-s + 1.52i·7-s + (0.448 + 0.893i)8-s + 0.441·9-s + (0.712 + 0.275i)10-s + 0.334i·11-s + (−0.888 − 0.807i)12-s + 0.0768i·13-s + (−0.548 + 1.42i)14-s − 0.916·15-s + (0.0961 + 0.995i)16-s + 1.43·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.112 - 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.112 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.112 - 0.993i$
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: \((2,\ 76,\ (\ :3/2),\ 0.112 - 0.993i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.41855 + 1.26664i\)
\(L(\frac12)\) \(\approx\) \(1.41855 + 1.26664i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.63 - 1.01i)T \)
19 \( 1 + (48.4 + 67.2i)T \)
good3 \( 1 + 6.23T + 27T^{2} \)
5 \( 1 - 8.54T + 125T^{2} \)
7 \( 1 - 28.2iT - 343T^{2} \)
11 \( 1 - 12.2iT - 1.33e3T^{2} \)
13 \( 1 - 3.60iT - 2.19e3T^{2} \)
17 \( 1 - 100.T + 4.91e3T^{2} \)
23 \( 1 + 158. iT - 1.21e4T^{2} \)
29 \( 1 + 151. iT - 2.43e4T^{2} \)
31 \( 1 - 123.T + 2.97e4T^{2} \)
37 \( 1 - 64.7iT - 5.06e4T^{2} \)
41 \( 1 - 374. iT - 6.89e4T^{2} \)
43 \( 1 + 211. iT - 7.95e4T^{2} \)
47 \( 1 - 101. iT - 1.03e5T^{2} \)
53 \( 1 + 450. iT - 1.48e5T^{2} \)
59 \( 1 - 115.T + 2.05e5T^{2} \)
61 \( 1 - 310.T + 2.26e5T^{2} \)
67 \( 1 - 1.07e3T + 3.00e5T^{2} \)
71 \( 1 + 493.T + 3.57e5T^{2} \)
73 \( 1 + 617.T + 3.89e5T^{2} \)
79 \( 1 - 275.T + 4.93e5T^{2} \)
83 \( 1 + 566. iT - 5.71e5T^{2} \)
89 \( 1 - 1.62e3iT - 7.04e5T^{2} \)
97 \( 1 + 1.57e3iT - 9.12e5T^{2} \)
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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

−14.34838048544560960318692768189, −12.95929086224358471804304255461, −12.14573316619741846962157405780, −11.43950286074263802365721421667, −10.01708703953652386030819033599, −8.379476421158965187973547396257, −6.49928312321288310776352329723, −5.79550039045856852776706086430, −4.87909351701048810477262372439, −2.48104744637462157915448681132, 1.16573356310429375151438993223, 3.71428251947603802139585430211, 5.28259231042281901913538748236, 6.13459769849993495896454301939, 7.40879082870082608620999807381, 9.990683982517456775007495480727, 10.57403791588601465597836034459, 11.59681871948360618586896314128, 12.65886562753498790078585850899, 13.74317543925241802134874361471

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