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Properties

Label 2-74-37.34-c1-0-0
Degree $2$
Conductor $74$
Sign $-0.173 - 0.984i$
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  + (0.766 + 0.642i)2-s + (−2.41 + 2.02i)3-s + (0.173 + 0.984i)4-s + (−1.43 − 0.524i)5-s − 3.14·6-s + (4.53 + 1.65i)7-s + (−0.500 + 0.866i)8-s + (1.20 − 6.81i)9-s + (−0.766 − 1.32i)10-s + (0.546 − 0.947i)11-s + (−2.41 − 2.02i)12-s + (0.307 + 1.74i)13-s + (2.41 + 4.17i)14-s + (4.53 − 1.65i)15-s + (−0.939 + 0.342i)16-s + (0.511 − 2.90i)17-s + ⋯
L(s)  = 1  + (0.541 + 0.454i)2-s + (−1.39 + 1.16i)3-s + (0.0868 + 0.492i)4-s + (−0.643 − 0.234i)5-s − 1.28·6-s + (1.71 + 0.623i)7-s + (−0.176 + 0.306i)8-s + (0.400 − 2.27i)9-s + (−0.242 − 0.419i)10-s + (0.164 − 0.285i)11-s + (−0.696 − 0.584i)12-s + (0.0852 + 0.483i)13-s + (0.644 + 1.11i)14-s + (1.17 − 0.426i)15-s + (−0.234 + 0.0855i)16-s + (0.124 − 0.703i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(0.590892\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :1/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.549294 + 0.654763i\)
\(L(\frac12)\) \(\approx\) \(0.549294 + 0.654763i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 - 0.642i)T \)
37 \( 1 + (-5.84 + 1.69i)T \)
good3 \( 1 + (2.41 - 2.02i)T + (0.520 - 2.95i)T^{2} \)
5 \( 1 + (1.43 + 0.524i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-4.53 - 1.65i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (-0.546 + 0.947i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.307 - 1.74i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-0.511 + 2.90i)T + (-15.9 - 5.81i)T^{2} \)
19 \( 1 + (0.632 - 0.530i)T + (3.29 - 18.7i)T^{2} \)
23 \( 1 + (-0.121 - 0.210i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.78 + 4.82i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.73T + 31T^{2} \)
41 \( 1 + (-0.505 - 2.86i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 - 4.37T + 43T^{2} \)
47 \( 1 + (1.13 + 1.96i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.77 - 2.10i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (8.29 - 3.01i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (2.04 + 11.5i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (8.44 + 3.07i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (0.933 - 0.783i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 - 8.79T + 73T^{2} \)
79 \( 1 + (2.02 + 0.736i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-0.884 + 5.01i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (8.43 - 3.07i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (2.17 + 3.77i)T + (-48.5 + 84.0i)T^{2} \)
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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

−15.12931801159570462519645522689, −14.20063450622417183979253465582, −12.22630103730621949583157127269, −11.57600496985856823501502746550, −10.96388257936693944884151713273, −9.240795973415347293017242486728, −7.86798632400340685792797214219, −6.05714197420939780390160784761, −4.97338589124328749649671196315, −4.21244474004040785437106630639, 1.48628071627190231624132091261, 4.46444877350296453794281036012, 5.64784939369976856170397718391, 7.13520937655010068884152449768, 7.993548946050369024663024827558, 10.69899250411732209496152492089, 11.13741165173208978953527849237, 12.02058390032188298817114344032, 12.89679581963278608390371199166, 14.06822256024370564219458015108

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