Properties

Label 2-76-76.15-c1-0-5
Degree $2$
Conductor $76$
Sign $0.839 + 0.543i$
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  + (−0.312 + 1.37i)2-s + (−0.547 − 3.10i)3-s + (−1.80 − 0.862i)4-s + (1.46 − 1.23i)5-s + (4.45 + 0.215i)6-s + (1.58 + 0.917i)7-s + (1.75 − 2.21i)8-s + (−6.53 + 2.38i)9-s + (1.23 + 2.40i)10-s + (−2.10 + 1.21i)11-s + (−1.69 + 6.08i)12-s + (2.49 + 0.440i)13-s + (−1.76 + 1.90i)14-s + (−4.62 − 3.88i)15-s + (2.51 + 3.11i)16-s + (0.818 + 0.297i)17-s + ⋯
L(s)  = 1  + (−0.220 + 0.975i)2-s + (−0.316 − 1.79i)3-s + (−0.902 − 0.431i)4-s + (0.655 − 0.550i)5-s + (1.81 + 0.0879i)6-s + (0.600 + 0.346i)7-s + (0.619 − 0.784i)8-s + (−2.17 + 0.793i)9-s + (0.391 + 0.761i)10-s + (−0.635 + 0.366i)11-s + (−0.487 + 1.75i)12-s + (0.692 + 0.122i)13-s + (−0.470 + 0.509i)14-s + (−1.19 − 1.00i)15-s + (0.628 + 0.777i)16-s + (0.198 + 0.0722i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 + 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.839 + 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.839 + 0.543i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1/2),\ 0.839 + 0.543i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.761990 - 0.225101i\)
\(L(\frac12)\) \(\approx\) \(0.761990 - 0.225101i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.312 - 1.37i)T \)
19 \( 1 + (-4.35 - 0.0369i)T \)
good3 \( 1 + (0.547 + 3.10i)T + (-2.81 + 1.02i)T^{2} \)
5 \( 1 + (-1.46 + 1.23i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-1.58 - 0.917i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.10 - 1.21i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.49 - 0.440i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (-0.818 - 0.297i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (4.69 - 5.59i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (0.967 + 2.65i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-2.07 + 3.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.48iT - 37T^{2} \)
41 \( 1 + (1.43 - 0.252i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (-5.26 - 6.27i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (0.855 + 2.34i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (1.18 - 1.41i)T + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (6.43 + 2.34i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (8.58 + 7.20i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (2.59 - 0.945i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-6.12 + 5.13i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-2.02 - 11.4i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-0.908 - 5.15i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (8.19 + 4.73i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.18 + 1.09i)T + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (2.11 - 5.81i)T + (-74.3 - 62.3i)T^{2} \)
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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

−14.05282683082582877258708902759, −13.49532848097851777539704139720, −12.62127278297792399949229490895, −11.40600212180341000246998303885, −9.538976156226998333027484819568, −8.185857720796239802180415839868, −7.51405262242820659611118673279, −6.08803531223674575594378867979, −5.33834178028767348443249002810, −1.56569846403822636688391946995, 3.07627921115643240427239682067, 4.45898219318946145642921758202, 5.67597484690463575688542454921, 8.279019890160977658441550036293, 9.461871347814430463461731793400, 10.52928766767504844706868415710, 10.73614761190690216916212699285, 11.98706812950668687146504067933, 13.80875545902263965918899237196, 14.39193447130835282230666393976

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