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Properties

Label 2-74-37.10-c3-0-7
Degree $2$
Conductor $74$
Sign $-0.819 + 0.572i$
Analytic cond. $4.36614$
Root an. cond. $2.08953$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)2-s + (−1.31 − 2.26i)3-s + (−1.99 − 3.46i)4-s + (0.0116 + 0.0201i)5-s − 5.24·6-s + (−10.2 − 17.8i)7-s − 7.99·8-s + (10.0 − 17.4i)9-s + 0.0464·10-s − 53.0·11-s + (−5.24 + 9.07i)12-s + (16.8 + 29.1i)13-s − 41.1·14-s + (0.0304 − 0.0527i)15-s + (−8 + 13.8i)16-s + (37.2 − 64.5i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.252 − 0.436i)3-s + (−0.249 − 0.433i)4-s + (0.00103 + 0.00179i)5-s − 0.356·6-s + (−0.555 − 0.962i)7-s − 0.353·8-s + (0.372 − 0.645i)9-s + 0.00146·10-s − 1.45·11-s + (−0.126 + 0.218i)12-s + (0.358 + 0.621i)13-s − 0.786·14-s + (0.000524 − 0.000907i)15-s + (−0.125 + 0.216i)16-s + (0.531 − 0.920i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 + 0.572i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.819 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.819 + 0.572i$
Analytic conductor: \(4.36614\)
Root analytic conductor: \(2.08953\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :3/2),\ -0.819 + 0.572i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.386197 - 1.22753i\)
\(L(\frac12)\) \(\approx\) \(0.386197 - 1.22753i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
37 \( 1 + (-201. + 100. i)T \)
good3 \( 1 + (1.31 + 2.26i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-0.0116 - 0.0201i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (10.2 + 17.8i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + 53.0T + 1.33e3T^{2} \)
13 \( 1 + (-16.8 - 29.1i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-37.2 + 64.5i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-9.14 - 15.8i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 - 193.T + 1.21e4T^{2} \)
29 \( 1 - 13.3T + 2.43e4T^{2} \)
31 \( 1 - 53.1T + 2.97e4T^{2} \)
41 \( 1 + (-182. - 315. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 - 81.2T + 7.95e4T^{2} \)
47 \( 1 + 533.T + 1.03e5T^{2} \)
53 \( 1 + (204. - 353. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-99.7 + 172. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (107. + 186. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (148. + 256. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (326. + 565. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 127.T + 3.89e5T^{2} \)
79 \( 1 + (132. + 230. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (383. - 664. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (97.3 - 168. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.03e3T + 9.12e5T^{2} \)
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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

−13.27001087498541824723060519377, −12.76928989133808371372286253979, −11.47573487362220349960273548534, −10.41889105458792105558615448038, −9.396707152525928052886681482638, −7.56129759161720422978932783565, −6.39973412429492422621157375978, −4.72668142312873806259463446499, −3.08767587022930525709162341520, −0.792571699756358183344262394559, 3.00492724898484373868975854519, 4.97607333819754380681812430234, 5.82050626877465117666162595346, 7.47842702295685821519682753384, 8.659521470073844708774739672184, 10.06135224808891995691193202490, 11.12785256947119723674654608639, 12.86211686503220656571317028674, 13.09795885506630677209769172749, 14.87750752977249975314306447888

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