Properties

Label 2-76-76.59-c1-0-6
Degree $2$
Conductor $76$
Sign $-0.364 + 0.931i$
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  + (0.343 − 1.37i)2-s + (−1.20 − 1.00i)3-s + (−1.76 − 0.942i)4-s + (0.709 − 0.258i)5-s + (−1.79 + 1.30i)6-s + (0.937 − 0.541i)7-s + (−1.89 + 2.09i)8-s + (−0.0947 − 0.537i)9-s + (−0.110 − 1.06i)10-s + (2.64 + 1.52i)11-s + (1.16 + 2.90i)12-s + (2.97 + 3.54i)13-s + (−0.420 − 1.47i)14-s + (−1.11 − 0.404i)15-s + (2.22 + 3.32i)16-s + (0.837 − 4.75i)17-s + ⋯
L(s)  = 1  + (0.242 − 0.970i)2-s + (−0.692 − 0.581i)3-s + (−0.882 − 0.471i)4-s + (0.317 − 0.115i)5-s + (−0.732 + 0.530i)6-s + (0.354 − 0.204i)7-s + (−0.671 + 0.741i)8-s + (−0.0315 − 0.179i)9-s + (−0.0349 − 0.335i)10-s + (0.798 + 0.460i)11-s + (0.337 + 0.839i)12-s + (0.825 + 0.983i)13-s + (−0.112 − 0.393i)14-s + (−0.287 − 0.104i)15-s + (0.556 + 0.831i)16-s + (0.203 − 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.364 + 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.364 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.364 + 0.931i$
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: \((2,\ 76,\ (\ :1/2),\ -0.364 + 0.931i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.487691 - 0.714797i\)
\(L(\frac12)\) \(\approx\) \(0.487691 - 0.714797i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.343 + 1.37i)T \)
19 \( 1 + (1.62 - 4.04i)T \)
good3 \( 1 + (1.20 + 1.00i)T + (0.520 + 2.95i)T^{2} \)
5 \( 1 + (-0.709 + 0.258i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (-0.937 + 0.541i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.64 - 1.52i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.97 - 3.54i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.837 + 4.75i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (-1.30 + 3.57i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (6.38 - 1.12i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (-0.249 - 0.431i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.70iT - 37T^{2} \)
41 \( 1 + (3.63 - 4.32i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (2.40 + 6.59i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (-8.31 + 1.46i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (-4.64 + 12.7i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (0.185 - 1.05i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-5.54 - 2.01i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-2.28 - 12.9i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (2.22 - 0.810i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-4.77 - 4.00i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (10.7 + 8.98i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (10.7 - 6.20i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.51 - 4.19i)T + (-15.4 + 87.6i)T^{2} \)
97 \( 1 + (3.44 + 0.608i)T + (91.1 + 33.1i)T^{2} \)
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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

−13.93078528788453513501373306904, −12.89998354544220871692156295299, −11.82780379078330814473891605295, −11.32482730361999588592887712651, −9.858247960635506189104743317721, −8.804853019493197274953259900737, −6.86034863068678348981853391220, −5.56704127536602940142815840573, −3.97762638362003162624142855196, −1.55610429297288547306017498367, 3.92361406170793012401336905893, 5.43445219243008973535123619138, 6.19592704352621259378863981378, 7.900713990501610196785364274902, 9.043825003596285347938283598827, 10.44065948921741653379421542315, 11.51005164611055255468392271471, 12.96848063694838041654303202953, 13.90146362388493177008688995230, 15.11332902831636362138337793568

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