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Properties

Label 2-74-37.6-c4-0-12
Degree $2$
Conductor $74$
Sign $-0.997 + 0.0666i$
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 − 8.34i·3-s − 8i·4-s + (−6.01 − 6.01i)5-s + (−16.6 − 16.6i)6-s − 74.3·7-s + (−16 − 16i)8-s + 11.3·9-s − 24.0·10-s − 29.8i·11-s − 66.7·12-s + (74.7 + 74.7i)13-s + (−148. + 148. i)14-s + (−50.1 + 50.1i)15-s − 64·16-s + (−128. − 128. i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s − 0.927i·3-s − 0.5i·4-s + (−0.240 − 0.240i)5-s + (−0.463 − 0.463i)6-s − 1.51·7-s + (−0.250 − 0.250i)8-s + 0.140·9-s − 0.240·10-s − 0.246i·11-s − 0.463·12-s + (0.442 + 0.442i)13-s + (−0.758 + 0.758i)14-s + (−0.222 + 0.222i)15-s − 0.250·16-s + (−0.444 − 0.444i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0666i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.997 + 0.0666i$
Analytic conductor: \(7.64937\)
Root analytic conductor: \(2.76575\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :2),\ -0.997 + 0.0666i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0453760 - 1.36068i\)
\(L(\frac12)\) \(\approx\) \(0.0453760 - 1.36068i\)
\(L(3)\) 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 \( 1 + (-1.35e3 + 203. i)T \)
good3 \( 1 + 8.34iT - 81T^{2} \)
5 \( 1 + (6.01 + 6.01i)T + 625iT^{2} \)
7 \( 1 + 74.3T + 2.40e3T^{2} \)
11 \( 1 + 29.8iT - 1.46e4T^{2} \)
13 \( 1 + (-74.7 - 74.7i)T + 2.85e4iT^{2} \)
17 \( 1 + (128. + 128. i)T + 8.35e4iT^{2} \)
19 \( 1 + (356. + 356. i)T + 1.30e5iT^{2} \)
23 \( 1 + (-274. - 274. i)T + 2.79e5iT^{2} \)
29 \( 1 + (-861. + 861. i)T - 7.07e5iT^{2} \)
31 \( 1 + (456. - 456. i)T - 9.23e5iT^{2} \)
41 \( 1 + 669. iT - 2.82e6T^{2} \)
43 \( 1 + (-116. - 116. i)T + 3.41e6iT^{2} \)
47 \( 1 - 1.80e3T + 4.87e6T^{2} \)
53 \( 1 - 275.T + 7.89e6T^{2} \)
59 \( 1 + (-444. - 444. i)T + 1.21e7iT^{2} \)
61 \( 1 + (-3.46e3 + 3.46e3i)T - 1.38e7iT^{2} \)
67 \( 1 - 2.63e3iT - 2.01e7T^{2} \)
71 \( 1 + 4.70e3T + 2.54e7T^{2} \)
73 \( 1 - 7.25e3iT - 2.83e7T^{2} \)
79 \( 1 + (6.84e3 + 6.84e3i)T + 3.89e7iT^{2} \)
83 \( 1 - 238.T + 4.74e7T^{2} \)
89 \( 1 + (1.03e4 - 1.03e4i)T - 6.27e7iT^{2} \)
97 \( 1 + (2.98e3 + 2.98e3i)T + 8.85e7iT^{2} \)
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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

−13.12034392672167077148287179728, −12.50298567897089435271286509812, −11.37288991854440752699601617938, −9.989930620261585373959139993612, −8.775950611221382035422076270839, −7.00536419520539993758341149518, −6.20420300610837543106316734308, −4.22539491976696010957428205553, −2.58019232609086302700436690434, −0.59540671638606188166765059378, 3.24661176368031213867758462575, 4.28844186061404941562378102349, 5.95435306926325769917666208641, 7.05887396664170236216905613251, 8.723805093264118052010330621221, 9.928452161205861826551959372251, 10.84370886892270427503147562165, 12.57023101403749878047092183959, 13.14678888780320000020955309679, 14.71683296694507803544111236959

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