Properties

Label 2-76-76.59-c1-0-1
Degree $2$
Conductor $76$
Sign $0.0902 - 0.995i$
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  + (−0.146 + 1.40i)2-s + (1.20 + 1.00i)3-s + (−1.95 − 0.412i)4-s + (0.709 − 0.258i)5-s + (−1.59 + 1.54i)6-s + (−0.937 + 0.541i)7-s + (0.866 − 2.69i)8-s + (−0.0947 − 0.537i)9-s + (0.259 + 1.03i)10-s + (−2.64 − 1.52i)11-s + (−1.93 − 2.46i)12-s + (2.97 + 3.54i)13-s + (−0.624 − 1.39i)14-s + (1.11 + 0.404i)15-s + (3.66 + 1.61i)16-s + (0.837 − 4.75i)17-s + ⋯
L(s)  = 1  + (−0.103 + 0.994i)2-s + (0.692 + 0.581i)3-s + (−0.978 − 0.206i)4-s + (0.317 − 0.115i)5-s + (−0.650 + 0.628i)6-s + (−0.354 + 0.204i)7-s + (0.306 − 0.951i)8-s + (−0.0315 − 0.179i)9-s + (0.0820 + 0.327i)10-s + (−0.798 − 0.460i)11-s + (−0.558 − 0.711i)12-s + (0.825 + 0.983i)13-s + (−0.166 − 0.373i)14-s + (0.287 + 0.104i)15-s + (0.915 + 0.403i)16-s + (0.203 − 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0902 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0902 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.0902 - 0.995i$
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: \((2,\ 76,\ (\ :1/2),\ 0.0902 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.731232 + 0.667948i\)
\(L(\frac12)\) \(\approx\) \(0.731232 + 0.667948i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.146 - 1.40i)T \)
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

−14.92187584057338931425450506533, −13.80887326188220737877592552927, −13.23653448004415687692540899683, −11.40214930081325635722250939155, −9.612381494870970173338897660633, −9.247872369671379469263250465543, −7.970263427864867504773412463442, −6.50812387634537586954372359575, −5.14294768992062653407364154922, −3.48065716452758005034365766710, 2.07652963311676610486041382551, 3.61301349341620332345831215365, 5.64958358190154016126938083436, 7.72590861513373747486267955359, 8.516650897769606244583690799066, 10.06598376664749676039648158561, 10.70618043597520280805054094755, 12.38519524314624761177672459300, 13.11934069652099949068923889505, 13.83776128719828355141772266594

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