Properties

Label 2-74-37.30-c1-0-0
Degree $2$
Conductor $74$
Sign $0.636 - 0.771i$
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 + (0.839 − 0.148i)5-s + 0.347i·6-s + (0.240 + 1.36i)7-s + (−0.866 − 0.500i)8-s + (−2.20 − 1.85i)9-s + (0.426 + 0.738i)10-s + (0.466 − 0.807i)11-s + (−0.326 + 0.118i)12-s + (−2.34 − 2.78i)13-s + (−1.19 + 0.692i)14-s + (0.291 + 0.0514i)15-s + (0.173 − 0.984i)16-s + (2.84 − 3.39i)17-s + ⋯
L(s)  = 1  + (0.241 + 0.664i)2-s + (0.188 + 0.0685i)3-s + (−0.383 + 0.321i)4-s + (0.375 − 0.0662i)5-s + 0.141i·6-s + (0.0908 + 0.515i)7-s + (−0.306 − 0.176i)8-s + (−0.735 − 0.616i)9-s + (0.134 + 0.233i)10-s + (0.140 − 0.243i)11-s + (−0.0942 + 0.0342i)12-s + (−0.649 − 0.773i)13-s + (−0.320 + 0.185i)14-s + (0.0752 + 0.0132i)15-s + (0.0434 − 0.246i)16-s + (0.690 − 0.822i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.636 - 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.636 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.636 - 0.771i$
Analytic conductor: \(0.590892\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :1/2),\ 0.636 - 0.771i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.957343 + 0.451318i\)
\(L(\frac12)\) \(\approx\) \(0.957343 + 0.451318i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.15 - 5.97i)T \)
good3 \( 1 + (-0.326 - 0.118i)T + (2.29 + 1.92i)T^{2} \)
5 \( 1 + (-0.839 + 0.148i)T + (4.69 - 1.71i)T^{2} \)
7 \( 1 + (-0.240 - 1.36i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (-0.466 + 0.807i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.34 + 2.78i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (-2.84 + 3.39i)T + (-2.95 - 16.7i)T^{2} \)
19 \( 1 + (1.30 - 3.59i)T + (-14.5 - 12.2i)T^{2} \)
23 \( 1 + (-0.920 + 0.531i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.873 - 0.504i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.33iT - 31T^{2} \)
41 \( 1 + (0.186 - 0.156i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 - 5.13iT - 43T^{2} \)
47 \( 1 + (-3.89 - 6.75i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.25 + 12.7i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (9.61 + 1.69i)T + (55.4 + 20.1i)T^{2} \)
61 \( 1 + (-0.255 - 0.304i)T + (-10.5 + 60.0i)T^{2} \)
67 \( 1 + (2.47 + 14.0i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-12.8 - 4.67i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + 13.1T + 73T^{2} \)
79 \( 1 + (-3.43 + 0.605i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (12.8 + 10.7i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (-6.19 - 1.09i)T + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (-6.47 + 3.73i)T + (48.5 - 84.0i)T^{2} \)
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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

−14.66881415660353042637821186834, −13.98384676084889000685589781391, −12.63473095090939545961575992414, −11.72734522133512595036373031581, −10.00147351595951351584387000234, −8.917788285720514549558166152836, −7.80346798402124169414641328814, −6.21786378394417933084008698964, −5.18612393138735563812583515025, −3.15142266030834501031636738799, 2.30041708353806212635810130905, 4.21432052587269823651922698858, 5.76299559646189544573590098072, 7.47088639121353521904711767998, 8.935182999307037125198464448526, 10.10553876755390729342022644947, 11.13153306834660212294718302589, 12.22237941282082469743621718439, 13.48508265528956214292706359324, 14.13495892563717408511072779660

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