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Properties

Label 2-74-37.12-c1-0-2
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: $\chi_{74} (49, \cdot )$
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

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

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