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Properties

Label 2-74-37.31-c6-0-6
Degree $2$
Conductor $74$
Sign $-0.889 - 0.457i$
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 + 27.1i·3-s + 32i·4-s + (−159. + 159. i)5-s + (108. − 108. i)6-s + 664.·7-s + (128 − 128i)8-s − 8.56·9-s + 1.27e3·10-s + 2.20e3i·11-s − 869.·12-s + (−473. + 473. i)13-s + (−2.65e3 − 2.65e3i)14-s + (−4.32e3 − 4.32e3i)15-s − 1.02e3·16-s + (1.34e3 − 1.34e3i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + 1.00i·3-s + 0.5i·4-s + (−1.27 + 1.27i)5-s + (0.502 − 0.502i)6-s + 1.93·7-s + (0.250 − 0.250i)8-s − 0.0117·9-s + 1.27·10-s + 1.65i·11-s − 0.502·12-s + (−0.215 + 0.215i)13-s + (−0.968 − 0.968i)14-s + (−1.28 − 1.28i)15-s − 0.250·16-s + (0.273 − 0.273i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 - 0.457i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.889 - 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.889 - 0.457i$
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: \((2,\ 74,\ (\ :3),\ -0.889 - 0.457i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.265090 + 1.09513i\)
\(L(\frac12)\) \(\approx\) \(0.265090 + 1.09513i\)
\(L(4)\) 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 \( 1 + (-4.29e4 - 2.68e4i)T \)
good3 \( 1 - 27.1iT - 729T^{2} \)
5 \( 1 + (159. - 159. i)T - 1.56e4iT^{2} \)
7 \( 1 - 664.T + 1.17e5T^{2} \)
11 \( 1 - 2.20e3iT - 1.77e6T^{2} \)
13 \( 1 + (473. - 473. i)T - 4.82e6iT^{2} \)
17 \( 1 + (-1.34e3 + 1.34e3i)T - 2.41e7iT^{2} \)
19 \( 1 + (506. - 506. i)T - 4.70e7iT^{2} \)
23 \( 1 + (1.02e4 - 1.02e4i)T - 1.48e8iT^{2} \)
29 \( 1 + (3.27e4 + 3.27e4i)T + 5.94e8iT^{2} \)
31 \( 1 + (-9.11e3 - 9.11e3i)T + 8.87e8iT^{2} \)
41 \( 1 + 6.73e4iT - 4.75e9T^{2} \)
43 \( 1 + (2.22e4 - 2.22e4i)T - 6.32e9iT^{2} \)
47 \( 1 - 5.39e4T + 1.07e10T^{2} \)
53 \( 1 + 4.37e4T + 2.21e10T^{2} \)
59 \( 1 + (4.43e4 - 4.43e4i)T - 4.21e10iT^{2} \)
61 \( 1 + (2.15e5 + 2.15e5i)T + 5.15e10iT^{2} \)
67 \( 1 + 1.58e5iT - 9.04e10T^{2} \)
71 \( 1 - 3.61e5T + 1.28e11T^{2} \)
73 \( 1 + 3.00e5iT - 1.51e11T^{2} \)
79 \( 1 + (3.70e5 - 3.70e5i)T - 2.43e11iT^{2} \)
83 \( 1 - 1.80e5T + 3.26e11T^{2} \)
89 \( 1 + (2.12e5 + 2.12e5i)T + 4.96e11iT^{2} \)
97 \( 1 + (3.37e5 - 3.37e5i)T - 8.32e11iT^{2} \)
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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

−14.24416843288706918318642064568, −12.06688095930827448140618430482, −11.40256652070639365696788359832, −10.59073025585604980188586389125, −9.622985131532112181657196736864, −7.87968006845668657440864543107, −7.38087397376586483913542591753, −4.67323284486632464514462355149, −3.89716323615498667272496991774, −2.02828075485298698607627716681, 0.56045951145769404544792973806, 1.45955608884057044693975700107, 4.34917298429743729177308715396, 5.61417520743173183722290153697, 7.52287974499117278336783063741, 8.136696279261392869315062688917, 8.703903917732885444171804204548, 10.99327063122569469024647589240, 11.75103786805024805999604790349, 12.80343423564609874898892060806

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