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Properties

Label 2-76-4.3-c4-0-1
Degree $2$
Conductor $76$
Sign $-0.976 + 0.214i$
Analytic cond. $7.85611$
Root an. cond. $2.80287$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.11 − 2.50i)2-s + 12.3i·3-s + (3.43 + 15.6i)4-s − 18.3·5-s + (30.8 − 38.3i)6-s + 80.8i·7-s + (28.4 − 57.3i)8-s − 70.7·9-s + (57.2 + 46.0i)10-s − 201. i·11-s + (−192. + 42.2i)12-s − 82.4·13-s + (202. − 251. i)14-s − 226. i·15-s + (−232. + 107. i)16-s − 145.·17-s + ⋯
L(s)  = 1  + (−0.779 − 0.626i)2-s + 1.36i·3-s + (0.214 + 0.976i)4-s − 0.734·5-s + (0.857 − 1.06i)6-s + 1.64i·7-s + (0.444 − 0.895i)8-s − 0.872·9-s + (0.572 + 0.460i)10-s − 1.66i·11-s + (−1.33 + 0.293i)12-s − 0.487·13-s + (1.03 − 1.28i)14-s − 1.00i·15-s + (−0.907 + 0.419i)16-s − 0.502·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.214i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.976 + 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.976 + 0.214i$
Analytic conductor: \(7.85611\)
Root analytic conductor: \(2.80287\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :2),\ -0.976 + 0.214i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0341008 - 0.314084i\)
\(L(\frac12)\) \(\approx\) \(0.0341008 - 0.314084i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.11 + 2.50i)T \)
19 \( 1 - 82.8iT \)
good3 \( 1 - 12.3iT - 81T^{2} \)
5 \( 1 + 18.3T + 625T^{2} \)
7 \( 1 - 80.8iT - 2.40e3T^{2} \)
11 \( 1 + 201. iT - 1.46e4T^{2} \)
13 \( 1 + 82.4T + 2.85e4T^{2} \)
17 \( 1 + 145.T + 8.35e4T^{2} \)
23 \( 1 - 87.6iT - 2.79e5T^{2} \)
29 \( 1 + 632.T + 7.07e5T^{2} \)
31 \( 1 + 1.53e3iT - 9.23e5T^{2} \)
37 \( 1 - 935.T + 1.87e6T^{2} \)
41 \( 1 - 536.T + 2.82e6T^{2} \)
43 \( 1 + 230. iT - 3.41e6T^{2} \)
47 \( 1 - 2.87e3iT - 4.87e6T^{2} \)
53 \( 1 + 4.52e3T + 7.89e6T^{2} \)
59 \( 1 + 785. iT - 1.21e7T^{2} \)
61 \( 1 - 4.08e3T + 1.38e7T^{2} \)
67 \( 1 - 7.03e3iT - 2.01e7T^{2} \)
71 \( 1 - 9.06e3iT - 2.54e7T^{2} \)
73 \( 1 + 6.79e3T + 2.83e7T^{2} \)
79 \( 1 - 7.15e3iT - 3.89e7T^{2} \)
83 \( 1 + 1.16e4iT - 4.74e7T^{2} \)
89 \( 1 + 4.82e3T + 6.27e7T^{2} \)
97 \( 1 + 7.04e3T + 8.85e7T^{2} \)
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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

−14.74125830883372150102038900730, −12.91629426402157134955169082969, −11.53176723856327248866187634668, −11.21998192463917193595053249844, −9.723869243849386853563834732918, −8.946295922216331391871401682070, −8.030241859637946319748980122721, −5.76836003942871631637519266700, −4.04974714726691340943893698155, −2.78948010870585453651865320056, 0.20583050435697222126903267224, 1.69671485976683336182950270560, 4.55032481532519533020904555467, 6.78114132691341292649355363769, 7.26750341046598756021911506166, 7.979713350438858354682711336013, 9.709263260817889284109008108895, 10.85466522022949894009916956110, 12.15038237396810169782852027118, 13.26253793549892861409186005532

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