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Properties

Label 2-74-37.31-c4-0-7
Degree $2$
Conductor $74$
Sign $0.485 + 0.874i$
Analytic cond. $7.64937$
Root an. cond. $2.76575$
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  + (−2 − 2i)2-s − 4.21i·3-s + 8i·4-s + (−19.1 + 19.1i)5-s + (−8.42 + 8.42i)6-s + 69.6·7-s + (16 − 16i)8-s + 63.2·9-s + 76.7·10-s − 199. i·11-s + 33.6·12-s + (−144. + 144. i)13-s + (−139. − 139. i)14-s + (80.7 + 80.7i)15-s − 64·16-s + (182. − 182. i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s − 0.468i·3-s + 0.5i·4-s + (−0.767 + 0.767i)5-s + (−0.234 + 0.234i)6-s + 1.42·7-s + (0.250 − 0.250i)8-s + 0.780·9-s + 0.767·10-s − 1.64i·11-s + 0.234·12-s + (−0.854 + 0.854i)13-s + (−0.710 − 0.710i)14-s + (0.359 + 0.359i)15-s − 0.250·16-s + (0.632 − 0.632i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.485 + 0.874i$
Analytic conductor: \(7.64937\)
Root analytic conductor: \(2.76575\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :2),\ 0.485 + 0.874i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.14633 - 0.674790i\)
\(L(\frac12)\) \(\approx\) \(1.14633 - 0.674790i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2 + 2i)T \)
37 \( 1 + (-563. - 1.24e3i)T \)
good3 \( 1 + 4.21iT - 81T^{2} \)
5 \( 1 + (19.1 - 19.1i)T - 625iT^{2} \)
7 \( 1 - 69.6T + 2.40e3T^{2} \)
11 \( 1 + 199. iT - 1.46e4T^{2} \)
13 \( 1 + (144. - 144. i)T - 2.85e4iT^{2} \)
17 \( 1 + (-182. + 182. i)T - 8.35e4iT^{2} \)
19 \( 1 + (-286. + 286. i)T - 1.30e5iT^{2} \)
23 \( 1 + (-522. + 522. i)T - 2.79e5iT^{2} \)
29 \( 1 + (-807. - 807. i)T + 7.07e5iT^{2} \)
31 \( 1 + (14.9 + 14.9i)T + 9.23e5iT^{2} \)
41 \( 1 + 1.99e3iT - 2.82e6T^{2} \)
43 \( 1 + (695. - 695. i)T - 3.41e6iT^{2} \)
47 \( 1 - 3.63e3T + 4.87e6T^{2} \)
53 \( 1 + 5.47e3T + 7.89e6T^{2} \)
59 \( 1 + (2.65e3 - 2.65e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (-2.14e3 - 2.14e3i)T + 1.38e7iT^{2} \)
67 \( 1 + 3.13e3iT - 2.01e7T^{2} \)
71 \( 1 + 8.72e3T + 2.54e7T^{2} \)
73 \( 1 + 2.58e3iT - 2.83e7T^{2} \)
79 \( 1 + (5.00e3 - 5.00e3i)T - 3.89e7iT^{2} \)
83 \( 1 - 7.65e3T + 4.74e7T^{2} \)
89 \( 1 + (-1.51e3 - 1.51e3i)T + 6.27e7iT^{2} \)
97 \( 1 + (-6.90e3 + 6.90e3i)T - 8.85e7iT^{2} \)
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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

−13.72605412421034479081209713508, −12.12192216965342467298607698532, −11.40092002327512275120799334804, −10.62207824480547383111683124808, −8.950394158147263154751661045345, −7.76924273187313265958156828034, −6.97786378336858502849589808210, −4.72180758482048889488858432193, −2.93936963553850864238812329000, −1.03119008537910743556548709971, 1.38520994873677481530998716518, 4.40185323385227281855998588852, 5.15622929895370431338305779798, 7.53177905565334774746745781722, 7.919996901897873182364757999028, 9.497033687113857615238904516394, 10.38522059964015699313010914383, 11.84936724820564927088262119746, 12.71660569510053623468212076283, 14.54627437617783770398698397304

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