Properties

Label 2-74-37.16-c3-0-0
Degree $2$
Conductor $74$
Sign $0.363 - 0.931i$
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 + (−1.18 + 0.429i)3-s + (3.06 + 2.57i)4-s + (0.00632 − 0.0358i)5-s + 2.51·6-s + (−0.240 + 1.36i)7-s + (−4.00 − 6.92i)8-s + (−19.4 + 16.3i)9-s + (−0.0363 + 0.0630i)10-s + (26.7 + 46.3i)11-s + (−4.72 − 1.71i)12-s + (58.2 + 48.8i)13-s + (1.38 − 2.39i)14-s + (0.00793 + 0.0450i)15-s + (2.77 + 15.7i)16-s + (−18.5 + 15.5i)17-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (−0.227 + 0.0826i)3-s + (0.383 + 0.321i)4-s + (0.000565 − 0.00320i)5-s + 0.170·6-s + (−0.0129 + 0.0735i)7-s + (−0.176 − 0.306i)8-s + (−0.721 + 0.605i)9-s + (−0.00115 + 0.00199i)10-s + (0.733 + 1.27i)11-s + (−0.113 − 0.0413i)12-s + (1.24 + 1.04i)13-s + (0.0263 − 0.0457i)14-s + (0.000136 + 0.000775i)15-s + (0.0434 + 0.246i)16-s + (−0.264 + 0.221i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.363 - 0.931i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.363 - 0.931i$
Analytic conductor: \(4.36614\)
Root analytic conductor: \(2.08953\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :3/2),\ 0.363 - 0.931i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.722600 + 0.493947i\)
\(L(\frac12)\) \(\approx\) \(0.722600 + 0.493947i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-203. + 96.5i)T \)
good3 \( 1 + (1.18 - 0.429i)T + (20.6 - 17.3i)T^{2} \)
5 \( 1 + (-0.00632 + 0.0358i)T + (-117. - 42.7i)T^{2} \)
7 \( 1 + (0.240 - 1.36i)T + (-322. - 117. i)T^{2} \)
11 \( 1 + (-26.7 - 46.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-58.2 - 48.8i)T + (381. + 2.16e3i)T^{2} \)
17 \( 1 + (18.5 - 15.5i)T + (853. - 4.83e3i)T^{2} \)
19 \( 1 + (10.5 - 3.84i)T + (5.25e3 - 4.40e3i)T^{2} \)
23 \( 1 + (71.2 - 123. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (145. + 251. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 150.T + 2.97e4T^{2} \)
41 \( 1 + (77.5 + 65.0i)T + (1.19e4 + 6.78e4i)T^{2} \)
43 \( 1 - 52.0T + 7.95e4T^{2} \)
47 \( 1 + (100. - 173. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (25.9 + 147. i)T + (-1.39e5 + 5.09e4i)T^{2} \)
59 \( 1 + (-3.38 - 19.2i)T + (-1.92e5 + 7.02e4i)T^{2} \)
61 \( 1 + (-156. - 131. i)T + (3.94e4 + 2.23e5i)T^{2} \)
67 \( 1 + (117. - 667. i)T + (-2.82e5 - 1.02e5i)T^{2} \)
71 \( 1 + (-487. + 177. i)T + (2.74e5 - 2.30e5i)T^{2} \)
73 \( 1 - 7.48T + 3.89e5T^{2} \)
79 \( 1 + (-95.0 + 539. i)T + (-4.63e5 - 1.68e5i)T^{2} \)
83 \( 1 + (1.08e3 - 913. i)T + (9.92e4 - 5.63e5i)T^{2} \)
89 \( 1 + (195. + 1.10e3i)T + (-6.62e5 + 2.41e5i)T^{2} \)
97 \( 1 + (-729. + 1.26e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−14.30754796278811391083588000332, −13.08487561497054992110686313956, −11.71333866558149384729105597577, −11.08567147646424795943279465504, −9.688312569779647246095028258497, −8.745332997394990761017814985769, −7.37004791821477491240364748969, −6.00031241631555899230313568510, −4.08790361830951441049291841289, −1.89106472678425080753079461426, 0.74652109679099088612404387382, 3.35352974824783660524768093422, 5.70840082937435669381802215213, 6.61584575594030909064220661511, 8.362823906695325356120361894763, 8.992522275627560603523787605507, 10.67186475323965347294008154059, 11.33469155209534560729595210799, 12.69743240313096063001901097709, 14.05285887674131722946557952691

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