Properties

Label 2-76-76.71-c1-0-7
Degree $2$
Conductor $76$
Sign $-0.471 + 0.881i$
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  + (−0.647 − 1.25i)2-s + (0.547 − 3.10i)3-s + (−1.16 + 1.62i)4-s + (1.46 + 1.23i)5-s + (−4.26 + 1.32i)6-s + (−1.58 + 0.917i)7-s + (2.79 + 0.408i)8-s + (−6.53 − 2.38i)9-s + (0.598 − 2.64i)10-s + (2.10 + 1.21i)11-s + (4.42 + 4.50i)12-s + (2.49 − 0.440i)13-s + (2.18 + 1.40i)14-s + (4.62 − 3.88i)15-s + (−1.29 − 3.78i)16-s + (0.818 − 0.297i)17-s + ⋯
L(s)  = 1  + (−0.457 − 0.889i)2-s + (0.316 − 1.79i)3-s + (−0.581 + 0.813i)4-s + (0.655 + 0.550i)5-s + (−1.74 + 0.539i)6-s + (−0.600 + 0.346i)7-s + (0.989 + 0.144i)8-s + (−2.17 − 0.793i)9-s + (0.189 − 0.834i)10-s + (0.635 + 0.366i)11-s + (1.27 + 1.30i)12-s + (0.692 − 0.122i)13-s + (0.583 + 0.375i)14-s + (1.19 − 1.00i)15-s + (−0.324 − 0.945i)16-s + (0.198 − 0.0722i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 + 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.471 + 0.881i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1/2),\ -0.471 + 0.881i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.424850 - 0.709130i\)
\(L(\frac12)\) \(\approx\) \(0.424850 - 0.709130i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.647 + 1.25i)T \)
19 \( 1 + (4.35 - 0.0369i)T \)
good3 \( 1 + (-0.547 + 3.10i)T + (-2.81 - 1.02i)T^{2} \)
5 \( 1 + (-1.46 - 1.23i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (1.58 - 0.917i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.10 - 1.21i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.49 + 0.440i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (-0.818 + 0.297i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (-4.69 - 5.59i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (0.967 - 2.65i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (2.07 + 3.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.48iT - 37T^{2} \)
41 \( 1 + (1.43 + 0.252i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (5.26 - 6.27i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (-0.855 + 2.34i)T + (-36.0 - 30.2i)T^{2} \)
53 \( 1 + (1.18 + 1.41i)T + (-9.20 + 52.1i)T^{2} \)
59 \( 1 + (-6.43 + 2.34i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (8.58 - 7.20i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-2.59 - 0.945i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (6.12 + 5.13i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-2.02 + 11.4i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (0.908 - 5.15i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-8.19 + 4.73i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.18 - 1.09i)T + (83.6 - 30.4i)T^{2} \)
97 \( 1 + (2.11 + 5.81i)T + (-74.3 + 62.3i)T^{2} \)
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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

−13.62170720854260973236423994382, −13.04152721046768570251116842993, −12.13918578591544475662460179002, −11.07665069438099097500600927347, −9.545418539713628666420275110275, −8.510900167216755830000101975005, −7.20310896585995174042526527490, −6.16244273790260290711291257881, −3.10062790110769999984955532617, −1.72706262235962652507823787028, 3.88574622954708484491988873326, 5.15622694516213646056835737151, 6.40108666389388120924604500671, 8.586134161987371752176027341782, 9.128337959861014492535173195638, 10.10475389036696841854948933848, 10.93482571954125291318103856411, 13.18560008124770770783219728002, 14.21804885856013311934604605706, 15.05658319069423123965215352474

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