Properties

Label 2-74-37.21-c1-0-0
Degree $2$
Conductor $74$
Sign $0.683 - 0.729i$
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  + (−0.342 + 0.939i)2-s + (0.326 − 0.118i)3-s + (−0.766 − 0.642i)4-s + (2.57 + 0.453i)5-s + 0.347i·6-s + (−0.361 + 2.04i)7-s + (0.866 − 0.500i)8-s + (−2.20 + 1.85i)9-s + (−1.30 + 2.26i)10-s + (−2.99 − 5.19i)11-s + (−0.326 − 0.118i)12-s + (2.64 − 3.15i)13-s + (−1.80 − 1.03i)14-s + (0.893 − 0.157i)15-s + (0.173 + 0.984i)16-s + (−0.618 − 0.737i)17-s + ⋯
L(s)  = 1  + (−0.241 + 0.664i)2-s + (0.188 − 0.0685i)3-s + (−0.383 − 0.321i)4-s + (1.15 + 0.202i)5-s + 0.141i·6-s + (−0.136 + 0.773i)7-s + (0.306 − 0.176i)8-s + (−0.735 + 0.616i)9-s + (−0.412 + 0.715i)10-s + (−0.903 − 1.56i)11-s + (−0.0942 − 0.0342i)12-s + (0.733 − 0.874i)13-s + (−0.481 − 0.277i)14-s + (0.230 − 0.0406i)15-s + (0.0434 + 0.246i)16-s + (−0.150 − 0.178i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.683 - 0.729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.683 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.683 - 0.729i$
Analytic conductor: \(0.590892\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :1/2),\ 0.683 - 0.729i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.842591 + 0.365173i\)
\(L(\frac12)\) \(\approx\) \(0.842591 + 0.365173i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.342 - 0.939i)T \)
37 \( 1 + (-6.07 - 0.383i)T \)
good3 \( 1 + (-0.326 + 0.118i)T + (2.29 - 1.92i)T^{2} \)
5 \( 1 + (-2.57 - 0.453i)T + (4.69 + 1.71i)T^{2} \)
7 \( 1 + (0.361 - 2.04i)T + (-6.57 - 2.39i)T^{2} \)
11 \( 1 + (2.99 + 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.64 + 3.15i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (0.618 + 0.737i)T + (-2.95 + 16.7i)T^{2} \)
19 \( 1 + (-0.534 - 1.46i)T + (-14.5 + 12.2i)T^{2} \)
23 \( 1 + (5.51 + 3.18i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.51 - 2.02i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.39iT - 31T^{2} \)
41 \( 1 + (-7.94 - 6.66i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 - 3.76iT - 43T^{2} \)
47 \( 1 + (-3.08 + 5.34i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.39 + 7.90i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (-5.02 + 0.885i)T + (55.4 - 20.1i)T^{2} \)
61 \( 1 + (6.25 - 7.45i)T + (-10.5 - 60.0i)T^{2} \)
67 \( 1 + (1.83 - 10.3i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (10.1 - 3.69i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 - 3.55T + 73T^{2} \)
79 \( 1 + (-2.51 - 0.442i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (-5.29 + 4.44i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (16.0 - 2.82i)T + (83.6 - 30.4i)T^{2} \)
97 \( 1 + (14.1 + 8.15i)T + (48.5 + 84.0i)T^{2} \)
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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

−14.61295628713703624918341568671, −13.73981841510577960413377916963, −13.03859763678394129810588830790, −11.15764598120730800311657781189, −10.15030202185912976102314786310, −8.771686294529636244560838956742, −8.035984789040003294085181929543, −5.93420191587257392533837633179, −5.65282805162515088236084507233, −2.74776678098280708047157717487, 2.13607921672580497508811577376, 4.16219124229754266687910926540, 5.92952947481102911955446889296, 7.57033113344179613471238587110, 9.216357267577790396664874122347, 9.817177415165528037415429575887, 10.97298420049924846231907112288, 12.31249833883572106235783952019, 13.40908159771612845373380319850, 14.05805349543429637179215936251

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