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Properties

Label 2-76-19.6-c1-0-0
Degree $2$
Conductor $76$
Sign $0.966 - 0.256i$
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  + (0.456 + 0.166i)3-s + (0.485 + 2.75i)5-s + (1.68 − 2.91i)7-s + (−2.11 − 1.77i)9-s + (0.258 + 0.447i)11-s + (−4.37 + 1.59i)13-s + (−0.235 + 1.33i)15-s + (−0.735 + 0.617i)17-s + (−3.12 − 3.04i)19-s + (1.25 − 1.05i)21-s + (0.629 − 3.57i)23-s + (−2.63 + 0.958i)25-s + (−1.40 − 2.42i)27-s + (6.21 + 5.21i)29-s + (−2.38 + 4.13i)31-s + ⋯
L(s)  = 1  + (0.263 + 0.0960i)3-s + (0.216 + 1.23i)5-s + (0.636 − 1.10i)7-s + (−0.705 − 0.592i)9-s + (0.0778 + 0.134i)11-s + (−1.21 + 0.441i)13-s + (−0.0609 + 0.345i)15-s + (−0.178 + 0.149i)17-s + (−0.715 − 0.698i)19-s + (0.273 − 0.229i)21-s + (0.131 − 0.744i)23-s + (−0.526 + 0.191i)25-s + (−0.269 − 0.467i)27-s + (1.15 + 0.968i)29-s + (−0.429 + 0.743i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.966 - 0.256i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1/2),\ 0.966 - 0.256i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00324 + 0.130669i\)
\(L(\frac12)\) \(\approx\) \(1.00324 + 0.130669i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (3.12 + 3.04i)T \)
good3 \( 1 + (-0.456 - 0.166i)T + (2.29 + 1.92i)T^{2} \)
5 \( 1 + (-0.485 - 2.75i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (-1.68 + 2.91i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.258 - 0.447i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.37 - 1.59i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (0.735 - 0.617i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (-0.629 + 3.57i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-6.21 - 5.21i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (2.38 - 4.13i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.13T + 37T^{2} \)
41 \( 1 + (6.54 + 2.38i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (0.817 + 4.63i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-10.4 - 8.74i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (1.20 - 6.81i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (10.9 - 9.19i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-1.05 + 6.00i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-3.38 - 2.84i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (1.66 + 9.42i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-5.65 - 2.05i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (9.85 + 3.58i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-2.39 + 4.14i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (8.30 - 3.02i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-8.46 + 7.10i)T + (16.8 - 95.5i)T^{2} \)
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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

−14.40976802076476336003589989946, −13.98299111261948096762027070493, −12.32317581276490526579771084095, −11.02212408631200887187996097394, −10.34028495820404225123472025460, −8.936519968234725533826720167201, −7.40473329084772466965499844888, −6.51722131074141707771854491724, −4.48779107652015503272330983523, −2.77055465366898505700181350683, 2.30298806306697667701588489468, 4.82408695978744130000053124508, 5.73500631592716291786890309492, 7.940815014043327935710845037687, 8.641974788424511633014989962291, 9.760269343573532536189195768520, 11.44640651642312856118774761003, 12.33085278406011538286051206712, 13.31380960018966064760246189585, 14.51893043166139748598860062107

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