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Properties

Label 2-76-76.75-c1-0-6
Degree $2$
Conductor $76$
Sign $0.508 + 0.860i$
Analytic cond. $0.606863$
Root an. cond. $0.779014$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.331 − 1.37i)2-s + 2.35·3-s + (−1.78 − 0.910i)4-s − 2.56·5-s + (0.780 − 3.24i)6-s + 4.15i·7-s + (−1.84 + 2.14i)8-s + 2.56·9-s + (−0.848 + 3.52i)10-s − 2.33i·11-s + (−4.19 − 2.14i)12-s − 4.29i·13-s + (5.70 + 1.37i)14-s − 6.04·15-s + (2.34 + 3.24i)16-s − 17-s + ⋯
L(s)  = 1  + (0.234 − 0.972i)2-s + 1.36·3-s + (−0.890 − 0.455i)4-s − 1.14·5-s + (0.318 − 1.32i)6-s + 1.56i·7-s + (−0.650 + 0.759i)8-s + 0.853·9-s + (−0.268 + 1.11i)10-s − 0.703i·11-s + (−1.21 − 0.619i)12-s − 1.19i·13-s + (1.52 + 0.367i)14-s − 1.55·15-s + (0.585 + 0.810i)16-s − 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 + 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.508 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.508 + 0.860i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1/2),\ 0.508 + 0.860i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02577 - 0.585356i\)
\(L(\frac12)\) \(\approx\) \(1.02577 - 0.585356i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.331 + 1.37i)T \)
19 \( 1 + (-3.68 - 2.33i)T \)
good3 \( 1 - 2.35T + 3T^{2} \)
5 \( 1 + 2.56T + 5T^{2} \)
7 \( 1 - 4.15iT - 7T^{2} \)
11 \( 1 + 2.33iT - 11T^{2} \)
13 \( 1 + 4.29iT - 13T^{2} \)
17 \( 1 + T + 17T^{2} \)
23 \( 1 + 1.82iT - 23T^{2} \)
29 \( 1 + 1.20iT - 29T^{2} \)
31 \( 1 + 1.32T + 31T^{2} \)
37 \( 1 - 5.49iT - 37T^{2} \)
41 \( 1 + 5.49iT - 41T^{2} \)
43 \( 1 - 1.30iT - 43T^{2} \)
47 \( 1 - 6.99iT - 47T^{2} \)
53 \( 1 + 9.79iT - 53T^{2} \)
59 \( 1 - 6.33T + 59T^{2} \)
61 \( 1 - 11.6T + 61T^{2} \)
67 \( 1 + 0.290T + 67T^{2} \)
71 \( 1 - 2.06T + 71T^{2} \)
73 \( 1 + 0.123T + 73T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 - 11.9iT - 83T^{2} \)
89 \( 1 - 14.0iT - 89T^{2} \)
97 \( 1 - 2.41iT - 97T^{2} \)
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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

−14.35805513301153449838420219782, −13.14041219027240167657850564905, −12.18144283195207747099499605380, −11.29093668359761925907665799131, −9.726777410095055001861729912255, −8.564326940179218256593446838710, −8.099422338167298163797584798119, −5.49894328770866825014555193549, −3.62028145778217057486891019273, −2.67371941033333324457382277444, 3.64174237733015602439561412964, 4.42148515038659471959616264176, 7.15159679734493210836232362493, 7.47962663616793606517488996346, 8.703488945975527412155544133601, 9.791228390138686301428530805517, 11.57505136425184562866212083122, 13.09101454595466601019012905474, 13.93133851744439537543564036865, 14.60006337977813350569178969393

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