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Properties

Label 2-76-76.75-c3-0-1
Degree $2$
Conductor $76$
Sign $-0.567 - 0.823i$
Analytic cond. $4.48414$
Root an. cond. $2.11758$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.78 − 0.515i)2-s − 3.18·3-s + (7.46 + 2.86i)4-s + 2.55·5-s + (8.84 + 1.64i)6-s − 16.6i·7-s + (−19.2 − 11.8i)8-s − 16.8·9-s + (−7.11 − 1.32i)10-s + 67.3i·11-s + (−23.7 − 9.12i)12-s + 61.8i·13-s + (−8.59 + 46.3i)14-s − 8.14·15-s + (47.5 + 42.8i)16-s − 83.0·17-s + ⋯
L(s)  = 1  + (−0.983 − 0.182i)2-s − 0.612·3-s + (0.933 + 0.358i)4-s + 0.228·5-s + (0.602 + 0.111i)6-s − 0.899i·7-s + (−0.852 − 0.522i)8-s − 0.625·9-s + (−0.225 − 0.0417i)10-s + 1.84i·11-s + (−0.571 − 0.219i)12-s + 1.32i·13-s + (−0.164 + 0.884i)14-s − 0.140·15-s + (0.742 + 0.669i)16-s − 1.18·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.567 - 0.823i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.567 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.567 - 0.823i$
Analytic conductor: \(4.48414\)
Root analytic conductor: \(2.11758\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :3/2),\ -0.567 - 0.823i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.146615 + 0.278936i\)
\(L(\frac12)\) \(\approx\) \(0.146615 + 0.278936i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.78 + 0.515i)T \)
19 \( 1 + (68.3 + 46.8i)T \)
good3 \( 1 + 3.18T + 27T^{2} \)
5 \( 1 - 2.55T + 125T^{2} \)
7 \( 1 + 16.6iT - 343T^{2} \)
11 \( 1 - 67.3iT - 1.33e3T^{2} \)
13 \( 1 - 61.8iT - 2.19e3T^{2} \)
17 \( 1 + 83.0T + 4.91e3T^{2} \)
23 \( 1 - 66.6iT - 1.21e4T^{2} \)
29 \( 1 - 136. iT - 2.43e4T^{2} \)
31 \( 1 - 81.2T + 2.97e4T^{2} \)
37 \( 1 + 227. iT - 5.06e4T^{2} \)
41 \( 1 - 414. iT - 6.89e4T^{2} \)
43 \( 1 + 166. iT - 7.95e4T^{2} \)
47 \( 1 - 158. iT - 1.03e5T^{2} \)
53 \( 1 + 557. iT - 1.48e5T^{2} \)
59 \( 1 + 397.T + 2.05e5T^{2} \)
61 \( 1 - 323.T + 2.26e5T^{2} \)
67 \( 1 + 143.T + 3.00e5T^{2} \)
71 \( 1 + 947.T + 3.57e5T^{2} \)
73 \( 1 - 253.T + 3.89e5T^{2} \)
79 \( 1 + 539.T + 4.93e5T^{2} \)
83 \( 1 - 196. iT - 5.71e5T^{2} \)
89 \( 1 + 931. iT - 7.04e5T^{2} \)
97 \( 1 + 432. iT - 9.12e5T^{2} \)
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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

−14.52437474764412471926058499966, −13.06618818498050718968751212996, −11.83265427333366644780245572266, −11.01526731535081129580324837897, −9.939866247402077063154136798964, −8.923819553747777911040253628881, −7.29151016503870548203140324517, −6.49323573098710876562844618270, −4.43438136227575893914216375389, −1.98211666490860867570668559097, 0.27466514376438485336447139300, 2.71358299782792923273153053035, 5.72426489165270730084292938331, 6.16474301320093788685921790096, 8.225921570624048745037414411194, 8.806888221657761440581452376395, 10.41330860082191751243434964286, 11.20265642399963425942130404614, 12.15627817918196937488912917347, 13.67931168680708648783620852149

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