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Properties

Label 2-74-37.31-c6-0-1
Degree $2$
Conductor $74$
Sign $-0.198 + 0.980i$
Analytic cond. $17.0240$
Root an. cond. $4.12601$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4 − 4i)2-s + 47.9i·3-s + 32i·4-s + (−25.3 + 25.3i)5-s + (191. − 191. i)6-s − 536.·7-s + (128 − 128i)8-s − 1.56e3·9-s + 203.·10-s + 1.61e3i·11-s − 1.53e3·12-s + (729. − 729. i)13-s + (2.14e3 + 2.14e3i)14-s + (−1.21e3 − 1.21e3i)15-s − 1.02e3·16-s + (5.90e3 − 5.90e3i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + 1.77i·3-s + 0.5i·4-s + (−0.203 + 0.203i)5-s + (0.887 − 0.887i)6-s − 1.56·7-s + (0.250 − 0.250i)8-s − 2.15·9-s + 0.203·10-s + 1.21i·11-s − 0.887·12-s + (0.332 − 0.332i)13-s + (0.781 + 0.781i)14-s + (−0.360 − 0.360i)15-s − 0.250·16-s + (1.20 − 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.198 + 0.980i$
Analytic conductor: \(17.0240\)
Root analytic conductor: \(4.12601\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :3),\ -0.198 + 0.980i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0339495 - 0.0415256i\)
\(L(\frac12)\) \(\approx\) \(0.0339495 - 0.0415256i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4 + 4i)T \)
37 \( 1 + (-1.41e4 + 4.86e4i)T \)
good3 \( 1 - 47.9iT - 729T^{2} \)
5 \( 1 + (25.3 - 25.3i)T - 1.56e4iT^{2} \)
7 \( 1 + 536.T + 1.17e5T^{2} \)
11 \( 1 - 1.61e3iT - 1.77e6T^{2} \)
13 \( 1 + (-729. + 729. i)T - 4.82e6iT^{2} \)
17 \( 1 + (-5.90e3 + 5.90e3i)T - 2.41e7iT^{2} \)
19 \( 1 + (-815. + 815. i)T - 4.70e7iT^{2} \)
23 \( 1 + (6.38e3 - 6.38e3i)T - 1.48e8iT^{2} \)
29 \( 1 + (-2.30e4 - 2.30e4i)T + 5.94e8iT^{2} \)
31 \( 1 + (3.02e4 + 3.02e4i)T + 8.87e8iT^{2} \)
41 \( 1 + 1.16e5iT - 4.75e9T^{2} \)
43 \( 1 + (-1.01e4 + 1.01e4i)T - 6.32e9iT^{2} \)
47 \( 1 + 1.74e5T + 1.07e10T^{2} \)
53 \( 1 - 3.44e4T + 2.21e10T^{2} \)
59 \( 1 + (2.46e4 - 2.46e4i)T - 4.21e10iT^{2} \)
61 \( 1 + (4.08e4 + 4.08e4i)T + 5.15e10iT^{2} \)
67 \( 1 + 8.02e3iT - 9.04e10T^{2} \)
71 \( 1 + 5.07e5T + 1.28e11T^{2} \)
73 \( 1 - 4.50e5iT - 1.51e11T^{2} \)
79 \( 1 + (3.82e5 - 3.82e5i)T - 2.43e11iT^{2} \)
83 \( 1 + 1.89e5T + 3.26e11T^{2} \)
89 \( 1 + (1.09e4 + 1.09e4i)T + 4.96e11iT^{2} \)
97 \( 1 + (-5.08e5 + 5.08e5i)T - 8.32e11iT^{2} \)
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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

−14.36330106027919194868458515807, −12.81460352004282817986970799024, −11.64376971402239798504749512085, −10.40641543675995089703685398865, −9.735117637909163229563461053096, −9.153733379128212611892642883863, −7.30306586012225528041145193937, −5.43414293247724852819836630719, −3.85071531437341339217354111586, −2.99863339365881656810824248407, 0.02681988416134481834529907792, 1.20969159619061877317343788043, 3.12142486474346388274874071146, 6.14673517192607166211618294461, 6.37535636315147773535060790341, 7.88657342196612171117837012036, 8.618780886623308032821168677776, 10.16551462719634856019919392032, 11.76257360939335551766155544813, 12.71791437678741047715061893169

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