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Properties

Label 2-76-76.75-c5-0-3
Degree $2$
Conductor $76$
Sign $0.758 - 0.652i$
Analytic cond. $12.1891$
Root an. cond. $3.49129$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.73 − 4.25i)2-s − 29.4·3-s + (−4.15 − 31.7i)4-s − 71.3·5-s + (−109. + 125. i)6-s − 160. i·7-s + (−150. − 100. i)8-s + 625.·9-s + (−266. + 303. i)10-s + 139. i·11-s + (122. + 935. i)12-s + 946. i·13-s + (−681. − 598. i)14-s + 2.10e3·15-s + (−989. + 263. i)16-s + 880.·17-s + ⋯
L(s)  = 1  + (0.659 − 0.751i)2-s − 1.89·3-s + (−0.129 − 0.991i)4-s − 1.27·5-s + (−1.24 + 1.42i)6-s − 1.23i·7-s + (−0.830 − 0.556i)8-s + 2.57·9-s + (−0.842 + 0.959i)10-s + 0.346i·11-s + (0.245 + 1.87i)12-s + 1.55i·13-s + (−0.929 − 0.815i)14-s + 2.41·15-s + (−0.966 + 0.257i)16-s + 0.738·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 - 0.652i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.758 - 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.758 - 0.652i$
Analytic conductor: \(12.1891\)
Root analytic conductor: \(3.49129\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :5/2),\ 0.758 - 0.652i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.249195 + 0.0924173i\)
\(L(\frac12)\) \(\approx\) \(0.249195 + 0.0924173i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.73 + 4.25i)T \)
19 \( 1 + (862. + 1.31e3i)T \)
good3 \( 1 + 29.4T + 243T^{2} \)
5 \( 1 + 71.3T + 3.12e3T^{2} \)
7 \( 1 + 160. iT - 1.68e4T^{2} \)
11 \( 1 - 139. iT - 1.61e5T^{2} \)
13 \( 1 - 946. iT - 3.71e5T^{2} \)
17 \( 1 - 880.T + 1.41e6T^{2} \)
23 \( 1 + 1.49e3iT - 6.43e6T^{2} \)
29 \( 1 + 2.44e3iT - 2.05e7T^{2} \)
31 \( 1 - 196.T + 2.86e7T^{2} \)
37 \( 1 + 4.50e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.59e4iT - 1.15e8T^{2} \)
43 \( 1 - 1.20e4iT - 1.47e8T^{2} \)
47 \( 1 - 2.36e4iT - 2.29e8T^{2} \)
53 \( 1 - 3.99e3iT - 4.18e8T^{2} \)
59 \( 1 - 2.89e4T + 7.14e8T^{2} \)
61 \( 1 - 1.13e4T + 8.44e8T^{2} \)
67 \( 1 + 2.74e4T + 1.35e9T^{2} \)
71 \( 1 - 6.68e3T + 1.80e9T^{2} \)
73 \( 1 + 8.13e4T + 2.07e9T^{2} \)
79 \( 1 - 1.25e4T + 3.07e9T^{2} \)
83 \( 1 - 1.81e4iT - 3.93e9T^{2} \)
89 \( 1 + 1.16e4iT - 5.58e9T^{2} \)
97 \( 1 - 3.11e4iT - 8.58e9T^{2} \)
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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

−13.21049411561172436568672664181, −12.21216160129909518362188125283, −11.44945912067288912263364344678, −10.90803178613383843518574094673, −9.801893208641640121996038576094, −7.29022022006470666650041122061, −6.36662829083839644975520644326, −4.57981608709590908475516422439, −4.17765794243129359406933354051, −1.05610571241298473035299273866, 0.14995409550618020448429661690, 3.69761881656195188757290093531, 5.28567195960990829252352081995, 5.80062716792123393613046601359, 7.23809036347785763232978795308, 8.357658910591016499425100366853, 10.46799163734462831637169279558, 11.77369298588620649892951016053, 12.11449244135231289970010341704, 12.95908258919980820274967429618

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