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Properties

Label 2-74-37.27-c3-0-6
Degree $2$
Conductor $74$
Sign $0.633 + 0.774i$
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  + (1.73 + i)2-s + (−2.54 − 4.40i)3-s + (1.99 + 3.46i)4-s + (9.68 − 5.59i)5-s − 10.1i·6-s + (−10.2 − 17.6i)7-s + 7.99i·8-s + (0.565 − 0.978i)9-s + 22.3·10-s + 15.2·11-s + (10.1 − 17.6i)12-s + (43.1 − 24.9i)13-s − 40.8i·14-s + (−49.2 − 28.4i)15-s + (−8 + 13.8i)16-s + (24.3 + 14.0i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.489 − 0.847i)3-s + (0.249 + 0.433i)4-s + (0.866 − 0.500i)5-s − 0.692i·6-s + (−0.551 − 0.954i)7-s + 0.353i·8-s + (0.0209 − 0.0362i)9-s + 0.707·10-s + 0.417·11-s + (0.244 − 0.423i)12-s + (0.920 − 0.531i)13-s − 0.779i·14-s + (−0.847 − 0.489i)15-s + (−0.125 + 0.216i)16-s + (0.347 + 0.200i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.633 + 0.774i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.633 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.633 + 0.774i$
Analytic conductor: \(4.36614\)
Root analytic conductor: \(2.08953\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :3/2),\ 0.633 + 0.774i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.74409 - 0.826817i\)
\(L(\frac12)\) \(\approx\) \(1.74409 - 0.826817i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.73 - i)T \)
37 \( 1 + (-224. - 15.1i)T \)
good3 \( 1 + (2.54 + 4.40i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-9.68 + 5.59i)T + (62.5 - 108. i)T^{2} \)
7 \( 1 + (10.2 + 17.6i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 - 15.2T + 1.33e3T^{2} \)
13 \( 1 + (-43.1 + 24.9i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-24.3 - 14.0i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (68.6 - 39.6i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 - 119. iT - 1.21e4T^{2} \)
29 \( 1 - 162. iT - 2.43e4T^{2} \)
31 \( 1 + 119. iT - 2.97e4T^{2} \)
41 \( 1 + (57.4 + 99.4i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 - 382. iT - 7.95e4T^{2} \)
47 \( 1 - 504.T + 1.03e5T^{2} \)
53 \( 1 + (36.3 - 62.9i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (24.4 + 14.1i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-71.0 + 41.0i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-87.2 - 151. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (514. + 890. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 32.7T + 3.89e5T^{2} \)
79 \( 1 + (461. - 266. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-590. + 1.02e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-1.31e3 - 756. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.75e3iT - 9.12e5T^{2} \)
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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

−13.41544447304793121237413039880, −13.19671089224111533231263261929, −12.12533992897764162737445618676, −10.73647757972315778200029664222, −9.362474741671031568599781811116, −7.70701842798012647462687616413, −6.48872205271282040147257888493, −5.71274709199986656646815979504, −3.81174215335531046426560663602, −1.28399509663510550758058348325, 2.41852608031923074921831684227, 4.20365603955815523861031068065, 5.69072951443446091031440877807, 6.49611496849524703876878948431, 8.943108921096314095047422630080, 10.00620783039098991623373891545, 10.85985361551298069620664589838, 11.95894377901576862936390431481, 13.18096933349074079481356786083, 14.18614957682922699976527364090

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