Properties

Label 2-76-76.51-c1-0-6
Degree $2$
Conductor $76$
Sign $0.755 + 0.655i$
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  + (0.719 − 1.21i)2-s + (1.09 + 0.397i)3-s + (−0.966 − 1.75i)4-s + (0.615 + 3.49i)5-s + (1.26 − 1.04i)6-s + (−3.53 − 2.04i)7-s + (−2.82 − 0.0826i)8-s + (−1.26 − 1.06i)9-s + (4.69 + 1.76i)10-s + (0.260 − 0.150i)11-s + (−0.358 − 2.29i)12-s + (0.546 + 1.50i)13-s + (−5.02 + 2.83i)14-s + (−0.715 + 4.05i)15-s + (−2.13 + 3.38i)16-s + (3.58 − 3.00i)17-s + ⋯
L(s)  = 1  + (0.508 − 0.861i)2-s + (0.629 + 0.229i)3-s + (−0.483 − 0.875i)4-s + (0.275 + 1.56i)5-s + (0.517 − 0.425i)6-s + (−1.33 − 0.771i)7-s + (−0.999 − 0.0292i)8-s + (−0.421 − 0.354i)9-s + (1.48 + 0.557i)10-s + (0.0784 − 0.0452i)11-s + (−0.103 − 0.662i)12-s + (0.151 + 0.416i)13-s + (−1.34 + 0.758i)14-s + (−0.184 + 1.04i)15-s + (−0.533 + 0.845i)16-s + (0.869 − 0.729i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.755 + 0.655i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1/2),\ 0.755 + 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14701 - 0.428161i\)
\(L(\frac12)\) \(\approx\) \(1.14701 - 0.428161i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.719 + 1.21i)T \)
19 \( 1 + (-3.34 - 2.78i)T \)
good3 \( 1 + (-1.09 - 0.397i)T + (2.29 + 1.92i)T^{2} \)
5 \( 1 + (-0.615 - 3.49i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (3.53 + 2.04i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.260 + 0.150i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.546 - 1.50i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-3.58 + 3.00i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (-3.07 - 0.541i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (-0.344 + 0.410i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (2.08 - 3.60i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.42iT - 37T^{2} \)
41 \( 1 + (-2.70 + 7.42i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (2.23 - 0.393i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (2.11 - 2.51i)T + (-8.16 - 46.2i)T^{2} \)
53 \( 1 + (5.12 + 0.903i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (4.17 - 3.50i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (0.594 - 3.37i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (5.29 + 4.44i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-0.743 - 4.21i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (8.50 + 3.09i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (0.187 + 0.0681i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-12.9 - 7.44i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.26 - 6.21i)T + (-68.1 + 57.2i)T^{2} \)
97 \( 1 + (-3.90 - 4.64i)T + (-16.8 + 95.5i)T^{2} \)
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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

−14.12512968434177919362565009980, −13.67093772246635880439902754491, −12.20393530620569858432990108897, −10.97172985890367242781759493109, −10.01079756597665898986560298128, −9.330520990161486662473278838817, −7.13098473459938365714599023387, −5.99398505926708027924370677873, −3.57328981837449753534348154315, −2.98615094361244422230341521145, 3.16753618430076477141385167648, 5.10394514549717531621415418465, 6.08567306646823718360710993726, 7.84644368519052779066034951358, 8.831271141356605739980877242498, 9.512302588254682447738467448979, 11.99220109450569392163671957279, 13.02913322366406677382551923401, 13.24675476914950867243412815266, 14.65591897459051028535311561543

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