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Properties

Label 2-74-37.6-c6-0-8
Degree $2$
Conductor $74$
Sign $0.127 + 0.991i$
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 − 20.1i·3-s − 32i·4-s + (9.27 + 9.27i)5-s + (80.7 + 80.7i)6-s + 8.30·7-s + (128 + 128i)8-s + 321.·9-s − 74.2·10-s − 243. i·11-s − 646.·12-s + (1.65e3 + 1.65e3i)13-s + (−33.2 + 33.2i)14-s + (187. − 187. i)15-s − 1.02e3·16-s + (−6.73e3 − 6.73e3i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s − 0.747i·3-s − 0.5i·4-s + (0.0742 + 0.0742i)5-s + (0.373 + 0.373i)6-s + 0.0242·7-s + (0.250 + 0.250i)8-s + 0.440·9-s − 0.0742·10-s − 0.183i·11-s − 0.373·12-s + (0.754 + 0.754i)13-s + (−0.0121 + 0.0121i)14-s + (0.0554 − 0.0554i)15-s − 0.250·16-s + (−1.37 − 1.37i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.127 + 0.991i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.127 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.127 + 0.991i$
Analytic conductor: \(17.0240\)
Root analytic conductor: \(4.12601\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :3),\ 0.127 + 0.991i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.914203 - 0.804100i\)
\(L(\frac12)\) \(\approx\) \(0.914203 - 0.804100i\)
\(L(4)\) 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 \( 1 + (1.05e4 + 4.95e4i)T \)
good3 \( 1 + 20.1iT - 729T^{2} \)
5 \( 1 + (-9.27 - 9.27i)T + 1.56e4iT^{2} \)
7 \( 1 - 8.30T + 1.17e5T^{2} \)
11 \( 1 + 243. iT - 1.77e6T^{2} \)
13 \( 1 + (-1.65e3 - 1.65e3i)T + 4.82e6iT^{2} \)
17 \( 1 + (6.73e3 + 6.73e3i)T + 2.41e7iT^{2} \)
19 \( 1 + (2.80e3 + 2.80e3i)T + 4.70e7iT^{2} \)
23 \( 1 + (-4.07e3 - 4.07e3i)T + 1.48e8iT^{2} \)
29 \( 1 + (-2.99e4 + 2.99e4i)T - 5.94e8iT^{2} \)
31 \( 1 + (4.34e3 - 4.34e3i)T - 8.87e8iT^{2} \)
41 \( 1 - 1.02e4iT - 4.75e9T^{2} \)
43 \( 1 + (7.68e4 + 7.68e4i)T + 6.32e9iT^{2} \)
47 \( 1 - 1.24e5T + 1.07e10T^{2} \)
53 \( 1 + 8.28e4T + 2.21e10T^{2} \)
59 \( 1 + (1.44e5 + 1.44e5i)T + 4.21e10iT^{2} \)
61 \( 1 + (7.04e4 - 7.04e4i)T - 5.15e10iT^{2} \)
67 \( 1 - 4.83e5iT - 9.04e10T^{2} \)
71 \( 1 - 1.72e5T + 1.28e11T^{2} \)
73 \( 1 + 5.26e5iT - 1.51e11T^{2} \)
79 \( 1 + (-6.16e5 - 6.16e5i)T + 2.43e11iT^{2} \)
83 \( 1 - 3.59e5T + 3.26e11T^{2} \)
89 \( 1 + (-3.12e5 + 3.12e5i)T - 4.96e11iT^{2} \)
97 \( 1 + (-3.44e4 - 3.44e4i)T + 8.32e11iT^{2} \)
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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

−13.40260329122090755212382645902, −11.96399092974041313984792182750, −10.85668390288155678617387825505, −9.459834418628746479864824931003, −8.399761765790040931540326593677, −7.07108962782857690326826020192, −6.36063578947337612749405393225, −4.53148258271448603935348572117, −2.16828170106321114567411814962, −0.57942719235115988273336425241, 1.50865653113639800937392322648, 3.43309086465723965571371415182, 4.70430530652631028438407710525, 6.53199306419323434980675469290, 8.197533834461638935067216809065, 9.177466456951153329184001055901, 10.44029668526375458923614332332, 10.91510814733648700747411925897, 12.52899803280226355545055292703, 13.30476212744030243587575813537

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