Properties

Label 2-76-76.59-c1-0-2
Degree $2$
Conductor $76$
Sign $0.901 + 0.433i$
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  + (−0.800 − 1.16i)2-s + (1.21 + 1.02i)3-s + (−0.718 + 1.86i)4-s + (2.19 − 0.798i)5-s + (0.216 − 2.23i)6-s + (−1.56 + 0.905i)7-s + (2.75 − 0.655i)8-s + (−0.0835 − 0.474i)9-s + (−2.68 − 1.91i)10-s + (−1.01 − 0.586i)11-s + (−2.77 + 1.53i)12-s + (−2.13 − 2.54i)13-s + (2.31 + 1.10i)14-s + (3.48 + 1.26i)15-s + (−2.96 − 2.68i)16-s + (−1.18 + 6.74i)17-s + ⋯
L(s)  = 1  + (−0.565 − 0.824i)2-s + (0.701 + 0.588i)3-s + (−0.359 + 0.933i)4-s + (0.981 − 0.357i)5-s + (0.0883 − 0.912i)6-s + (−0.593 + 0.342i)7-s + (0.972 − 0.231i)8-s + (−0.0278 − 0.158i)9-s + (−0.850 − 0.607i)10-s + (−0.306 − 0.176i)11-s + (−0.801 + 0.443i)12-s + (−0.591 − 0.704i)13-s + (0.617 + 0.295i)14-s + (0.899 + 0.327i)15-s + (−0.741 − 0.670i)16-s + (−0.288 + 1.63i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 + 0.433i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.901 + 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.901 + 0.433i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1/2),\ 0.901 + 0.433i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.871981 - 0.198922i\)
\(L(\frac12)\) \(\approx\) \(0.871981 - 0.198922i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.800 + 1.16i)T \)
19 \( 1 + (2.39 - 3.64i)T \)
good3 \( 1 + (-1.21 - 1.02i)T + (0.520 + 2.95i)T^{2} \)
5 \( 1 + (-2.19 + 0.798i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (1.56 - 0.905i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.01 + 0.586i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.13 + 2.54i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (1.18 - 6.74i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (-1.21 + 3.32i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.974 - 0.171i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (5.07 + 8.78i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.26iT - 37T^{2} \)
41 \( 1 + (-6.74 + 8.03i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (-2.85 - 7.83i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (-6.49 + 1.14i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (0.859 - 2.36i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (0.0105 - 0.0596i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-2.43 - 0.887i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-0.731 - 4.14i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-14.6 + 5.32i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-5.56 - 4.66i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (2.92 + 2.45i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (7.29 - 4.20i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.45 + 5.31i)T + (-15.4 + 87.6i)T^{2} \)
97 \( 1 + (-9.53 - 1.68i)T + (91.1 + 33.1i)T^{2} \)
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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

−14.38511544397075865241159814025, −13.00002194303227768072499605990, −12.54060257851516927701613986333, −10.72003373992897548636582513133, −9.846310639404854818491804767634, −9.106561553163136820843184757342, −8.077303680579530211011557089163, −5.94581923075025526079999530819, −3.94113873222161072322377100891, −2.41875925243608617176042322461, 2.32820088326045019862488850381, 5.14116342442512256640866077235, 6.76943324381202753985908667710, 7.38551739844580352841927990059, 8.981967044246494341022224296573, 9.708311694149606723918270858114, 10.94849810752847678696932158810, 12.97409273860328384617255339812, 13.85878863280714755906988065323, 14.31449543733849385830699519946

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