Properties

Label 2-74-37.33-c1-0-0
Degree $2$
Conductor $74$
Sign $-0.0933 - 0.995i$
Analytic cond. $0.590892$
Root an. cond. $0.768695$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.173 + 0.984i)2-s + (0.266 + 1.50i)3-s + (−0.939 − 0.342i)4-s + (−1.79 + 1.50i)5-s − 1.53·6-s + (1.93 − 1.62i)7-s + (0.5 − 0.866i)8-s + (0.613 − 0.223i)9-s + (−1.17 − 2.03i)10-s + (−0.560 + 0.970i)11-s + (0.266 − 1.50i)12-s + (1.70 + 0.620i)13-s + (1.26 + 2.19i)14-s + (−2.75 − 2.31i)15-s + (0.766 + 0.642i)16-s + (2.81 − 1.02i)17-s + ⋯
L(s)  = 1  + (−0.122 + 0.696i)2-s + (0.153 + 0.871i)3-s + (−0.469 − 0.171i)4-s + (−0.804 + 0.674i)5-s − 0.625·6-s + (0.733 − 0.615i)7-s + (0.176 − 0.306i)8-s + (0.204 − 0.0744i)9-s + (−0.371 − 0.642i)10-s + (−0.168 + 0.292i)11-s + (0.0768 − 0.435i)12-s + (0.473 + 0.172i)13-s + (0.338 + 0.586i)14-s + (−0.711 − 0.596i)15-s + (0.191 + 0.160i)16-s + (0.683 − 0.248i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0933 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0933 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.0933 - 0.995i$
Analytic conductor: \(0.590892\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :1/2),\ -0.0933 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.577433 + 0.634087i\)
\(L(\frac12)\) \(\approx\) \(0.577433 + 0.634087i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.173 - 0.984i)T \)
37 \( 1 + (-2.33 + 5.61i)T \)
good3 \( 1 + (-0.266 - 1.50i)T + (-2.81 + 1.02i)T^{2} \)
5 \( 1 + (1.79 - 1.50i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-1.93 + 1.62i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (0.560 - 0.970i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.70 - 0.620i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-2.81 + 1.02i)T + (13.0 - 10.9i)T^{2} \)
19 \( 1 + (0.971 + 5.51i)T + (-17.8 + 6.49i)T^{2} \)
23 \( 1 + (4.55 + 7.88i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.52 - 7.83i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.53T + 31T^{2} \)
41 \( 1 + (-6.98 - 2.54i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + 8.92T + 43T^{2} \)
47 \( 1 + (-0.194 - 0.337i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (9.23 + 7.74i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-6.41 - 5.38i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-2.45 - 0.892i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-5.62 + 4.72i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (0.448 + 2.54i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + 0.709T + 73T^{2} \)
79 \( 1 + (-3.39 + 2.84i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-6.81 + 2.47i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (9.95 + 8.35i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-7.34 - 12.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.79126996075159091001289688913, −14.44345191612597347524410721406, −12.85159607982860053185772858297, −11.20092133269274967520891102133, −10.45818334099382687659737312304, −9.136971278839220371114429063825, −7.81507301990169446241714833891, −6.84975231250723620386439949545, −4.85644310162187915331938115454, −3.76372049722118746170289440162, 1.67447562853816073528053822664, 3.93218650718567271627324299217, 5.66185389277856697238812700976, 7.898322386672053572906011818409, 8.174048153995259133981210448840, 9.806356439373998621442636560577, 11.37582881874450010097122845910, 12.10383584860166847713992591356, 12.93741711676873873325698142158, 13.98948526492443893576606521592

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