Properties

Label 2-74-37.7-c3-0-5
Degree $2$
Conductor $74$
Sign $0.988 + 0.149i$
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  + (1.87 − 0.684i)2-s + (6.85 + 2.49i)3-s + (3.06 − 2.57i)4-s + (−0.811 − 4.60i)5-s + 14.5·6-s + (−0.602 − 3.41i)7-s + (4.00 − 6.92i)8-s + (20.0 + 16.8i)9-s + (−4.67 − 8.09i)10-s + (−13.1 + 22.7i)11-s + (27.4 − 9.97i)12-s + (−28.7 + 24.1i)13-s + (−3.46 − 6.00i)14-s + (5.91 − 33.5i)15-s + (2.77 − 15.7i)16-s + (8.68 + 7.29i)17-s + ⋯
L(s)  = 1  + (0.664 − 0.241i)2-s + (1.31 + 0.479i)3-s + (0.383 − 0.321i)4-s + (−0.0726 − 0.411i)5-s + 0.992·6-s + (−0.0325 − 0.184i)7-s + (0.176 − 0.306i)8-s + (0.742 + 0.622i)9-s + (−0.147 − 0.256i)10-s + (−0.359 + 0.622i)11-s + (0.659 − 0.239i)12-s + (−0.614 + 0.515i)13-s + (−0.0661 − 0.114i)14-s + (0.101 − 0.577i)15-s + (0.0434 − 0.246i)16-s + (0.123 + 0.104i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.149i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.988 + 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.988 + 0.149i$
Analytic conductor: \(4.36614\)
Root analytic conductor: \(2.08953\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :3/2),\ 0.988 + 0.149i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.87271 - 0.216355i\)
\(L(\frac12)\) \(\approx\) \(2.87271 - 0.216355i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.87 + 0.684i)T \)
37 \( 1 + (185. - 128. i)T \)
good3 \( 1 + (-6.85 - 2.49i)T + (20.6 + 17.3i)T^{2} \)
5 \( 1 + (0.811 + 4.60i)T + (-117. + 42.7i)T^{2} \)
7 \( 1 + (0.602 + 3.41i)T + (-322. + 117. i)T^{2} \)
11 \( 1 + (13.1 - 22.7i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (28.7 - 24.1i)T + (381. - 2.16e3i)T^{2} \)
17 \( 1 + (-8.68 - 7.29i)T + (853. + 4.83e3i)T^{2} \)
19 \( 1 + (2.89 + 1.05i)T + (5.25e3 + 4.40e3i)T^{2} \)
23 \( 1 + (59.2 + 102. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (105. - 183. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 114.T + 2.97e4T^{2} \)
41 \( 1 + (0.174 - 0.146i)T + (1.19e4 - 6.78e4i)T^{2} \)
43 \( 1 - 176.T + 7.95e4T^{2} \)
47 \( 1 + (-7.57 - 13.1i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-107. + 607. i)T + (-1.39e5 - 5.09e4i)T^{2} \)
59 \( 1 + (78.1 - 443. i)T + (-1.92e5 - 7.02e4i)T^{2} \)
61 \( 1 + (-409. + 343. i)T + (3.94e4 - 2.23e5i)T^{2} \)
67 \( 1 + (-34.0 - 193. i)T + (-2.82e5 + 1.02e5i)T^{2} \)
71 \( 1 + (-564. - 205. i)T + (2.74e5 + 2.30e5i)T^{2} \)
73 \( 1 - 819.T + 3.89e5T^{2} \)
79 \( 1 + (-30.8 - 174. i)T + (-4.63e5 + 1.68e5i)T^{2} \)
83 \( 1 + (-100. - 84.1i)T + (9.92e4 + 5.63e5i)T^{2} \)
89 \( 1 + (-92.7 + 526. i)T + (-6.62e5 - 2.41e5i)T^{2} \)
97 \( 1 + (-759. - 1.31e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−14.26066707510474499301309037053, −13.07376256733467279418404491302, −12.17813998237154266004164599810, −10.56548903550688782866758860913, −9.521004061781664581406101714243, −8.420852830693115734326036144692, −7.04511893252912371992425953836, −4.98366580723744215948913154110, −3.78988763302025448829550882141, −2.28289461952286478078846827693, 2.42347440345264998547066957039, 3.57747021815766480492899964197, 5.59985437920868564449942891716, 7.24280761826284778296337013875, 8.037180277316153924002677258107, 9.320043937375822254676061650457, 10.88280801920275137116490274878, 12.29723140573124498217825626644, 13.32494264639596848187553471097, 14.06706400889343017408674864792

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