Properties

Label 2-74-37.16-c3-0-8
Degree $2$
Conductor $74$
Sign $0.909 + 0.414i$
Analytic cond. $4.36614$
Root an. cond. $2.08953$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.87 + 0.684i)2-s + (2.73 − 0.996i)3-s + (3.06 + 2.57i)4-s + (3.22 − 18.3i)5-s + 5.82·6-s + (2.30 − 13.0i)7-s + (4.00 + 6.92i)8-s + (−14.1 + 11.8i)9-s + (18.5 − 32.2i)10-s + (18.0 + 31.2i)11-s + (10.9 + 3.98i)12-s + (31.4 + 26.4i)13-s + (13.2 − 23.0i)14-s + (−9.41 − 53.3i)15-s + (2.77 + 15.7i)16-s + (−49.2 + 41.3i)17-s + ⋯
L(s)  = 1  + (0.664 + 0.241i)2-s + (0.527 − 0.191i)3-s + (0.383 + 0.321i)4-s + (0.288 − 1.63i)5-s + 0.396·6-s + (0.124 − 0.706i)7-s + (0.176 + 0.306i)8-s + (−0.525 + 0.440i)9-s + (0.588 − 1.01i)10-s + (0.494 + 0.856i)11-s + (0.263 + 0.0959i)12-s + (0.671 + 0.563i)13-s + (0.253 − 0.439i)14-s + (−0.161 − 0.918i)15-s + (0.0434 + 0.246i)16-s + (−0.703 + 0.590i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.414i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.909 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.909 + 0.414i$
Analytic conductor: \(4.36614\)
Root analytic conductor: \(2.08953\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :3/2),\ 0.909 + 0.414i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.44650 - 0.531173i\)
\(L(\frac12)\) \(\approx\) \(2.44650 - 0.531173i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.87 - 0.684i)T \)
37 \( 1 + (85.0 + 208. i)T \)
good3 \( 1 + (-2.73 + 0.996i)T + (20.6 - 17.3i)T^{2} \)
5 \( 1 + (-3.22 + 18.3i)T + (-117. - 42.7i)T^{2} \)
7 \( 1 + (-2.30 + 13.0i)T + (-322. - 117. i)T^{2} \)
11 \( 1 + (-18.0 - 31.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-31.4 - 26.4i)T + (381. + 2.16e3i)T^{2} \)
17 \( 1 + (49.2 - 41.3i)T + (853. - 4.83e3i)T^{2} \)
19 \( 1 + (15.2 - 5.56i)T + (5.25e3 - 4.40e3i)T^{2} \)
23 \( 1 + (-52.3 + 90.7i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-128. - 221. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 111.T + 2.97e4T^{2} \)
41 \( 1 + (78.9 + 66.2i)T + (1.19e4 + 6.78e4i)T^{2} \)
43 \( 1 + 271.T + 7.95e4T^{2} \)
47 \( 1 + (24.2 - 42.0i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (67.3 + 381. i)T + (-1.39e5 + 5.09e4i)T^{2} \)
59 \( 1 + (-54.5 - 309. i)T + (-1.92e5 + 7.02e4i)T^{2} \)
61 \( 1 + (-454. - 381. i)T + (3.94e4 + 2.23e5i)T^{2} \)
67 \( 1 + (-165. + 936. i)T + (-2.82e5 - 1.02e5i)T^{2} \)
71 \( 1 + (613. - 223. i)T + (2.74e5 - 2.30e5i)T^{2} \)
73 \( 1 - 962.T + 3.89e5T^{2} \)
79 \( 1 + (-200. + 1.13e3i)T + (-4.63e5 - 1.68e5i)T^{2} \)
83 \( 1 + (674. - 565. i)T + (9.92e4 - 5.63e5i)T^{2} \)
89 \( 1 + (-84.0 - 476. i)T + (-6.62e5 + 2.41e5i)T^{2} \)
97 \( 1 + (810. - 1.40e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−13.84835158692650832121862297824, −13.07651877070698880295115227321, −12.25408040786301011041171315218, −10.80400035437808292925459166232, −9.058647874839346280202647551923, −8.337350765919686201101683859504, −6.82300462808547688578958737738, −5.16199406069028225482212461248, −4.09609630261141830269807182304, −1.71684396218126337594150956042, 2.63038884760902630606136665917, 3.51556242299771364544181675484, 5.78174062851161240318751301614, 6.69959001601180411201026702336, 8.462163992884352545946230437642, 9.788586891222191756307846267478, 11.10964837754743341392110497323, 11.67718610042480289215518102703, 13.47895466686979550481564191829, 14.11401526606316381340630049151

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