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Properties

Label 2-76-76.75-c3-0-3
Degree $2$
Conductor $76$
Sign $0.161 - 0.986i$
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  + (−0.929 − 2.67i)2-s + 0.246·3-s + (−6.27 + 4.96i)4-s − 6.75·5-s + (−0.229 − 0.658i)6-s + 18.0i·7-s + (19.1 + 12.1i)8-s − 26.9·9-s + (6.27 + 18.0i)10-s + 27.9i·11-s + (−1.54 + 1.22i)12-s + 12.9i·13-s + (48.1 − 16.7i)14-s − 1.66·15-s + (14.6 − 62.3i)16-s + 37.2·17-s + ⋯
L(s)  = 1  + (−0.328 − 0.944i)2-s + 0.0474·3-s + (−0.783 + 0.621i)4-s − 0.603·5-s + (−0.0156 − 0.0448i)6-s + 0.973i·7-s + (0.844 + 0.536i)8-s − 0.997·9-s + (0.198 + 0.570i)10-s + 0.765i·11-s + (−0.0371 + 0.0294i)12-s + 0.277i·13-s + (0.919 − 0.320i)14-s − 0.0286·15-s + (0.228 − 0.973i)16-s + 0.531·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.161 - 0.986i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.161 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.161 - 0.986i$
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: \((2,\ 76,\ (\ :3/2),\ 0.161 - 0.986i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.397118 + 0.337396i\)
\(L(\frac12)\) \(\approx\) \(0.397118 + 0.337396i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.929 + 2.67i)T \)
19 \( 1 + (61.2 - 55.7i)T \)
good3 \( 1 - 0.246T + 27T^{2} \)
5 \( 1 + 6.75T + 125T^{2} \)
7 \( 1 - 18.0iT - 343T^{2} \)
11 \( 1 - 27.9iT - 1.33e3T^{2} \)
13 \( 1 - 12.9iT - 2.19e3T^{2} \)
17 \( 1 - 37.2T + 4.91e3T^{2} \)
23 \( 1 - 57.6iT - 1.21e4T^{2} \)
29 \( 1 + 132. iT - 2.43e4T^{2} \)
31 \( 1 + 246.T + 2.97e4T^{2} \)
37 \( 1 + 240. iT - 5.06e4T^{2} \)
41 \( 1 - 83.9iT - 6.89e4T^{2} \)
43 \( 1 - 267. iT - 7.95e4T^{2} \)
47 \( 1 + 221. iT - 1.03e5T^{2} \)
53 \( 1 - 617. iT - 1.48e5T^{2} \)
59 \( 1 - 827.T + 2.05e5T^{2} \)
61 \( 1 + 13.9T + 2.26e5T^{2} \)
67 \( 1 - 524.T + 3.00e5T^{2} \)
71 \( 1 - 460.T + 3.57e5T^{2} \)
73 \( 1 + 56.8T + 3.89e5T^{2} \)
79 \( 1 + 760.T + 4.93e5T^{2} \)
83 \( 1 - 574. iT - 5.71e5T^{2} \)
89 \( 1 - 811. iT - 7.04e5T^{2} \)
97 \( 1 + 1.51e3iT - 9.12e5T^{2} \)
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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

−14.20984209550757277831805969325, −12.75063224089111080798569791784, −11.95055683558365016886912956842, −11.17968979308812546049628642887, −9.749565015364523552457488919957, −8.724355554197676394050111541611, −7.70321997786894747874443930560, −5.57690274078292504162372057527, −3.87184655572036705891850125183, −2.26136917706687760778970731598, 0.36120353673320398219686110856, 3.75169368715408733757619321574, 5.38292374190285293470072379301, 6.79324985234146512774299438762, 7.968868355983659378600531062066, 8.837378843929647523625858348412, 10.36346275208117731456482184507, 11.33499108844234577732642226485, 13.01722516186408842363440004833, 14.06403054581458006884020354362

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