Properties

Label 2-74-37.16-c3-0-3
Degree $2$
Conductor $74$
Sign $0.615 - 0.787i$
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 + (6.42 − 2.33i)3-s + (3.06 + 2.57i)4-s + (−3.55 + 20.1i)5-s − 13.6·6-s + (−4.31 + 24.4i)7-s + (−4.00 − 6.92i)8-s + (15.1 − 12.6i)9-s + (20.4 − 35.4i)10-s + (−11.5 − 19.9i)11-s + (25.6 + 9.35i)12-s + (11.6 + 9.73i)13-s + (24.8 − 43.0i)14-s + (24.3 + 137. i)15-s + (2.77 + 15.7i)16-s + (−22.3 + 18.7i)17-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (1.23 − 0.449i)3-s + (0.383 + 0.321i)4-s + (−0.318 + 1.80i)5-s − 0.930·6-s + (−0.233 + 1.32i)7-s + (−0.176 − 0.306i)8-s + (0.559 − 0.469i)9-s + (0.648 − 1.12i)10-s + (−0.315 − 0.546i)11-s + (0.618 + 0.224i)12-s + (0.247 + 0.207i)13-s + (0.474 − 0.822i)14-s + (0.418 + 2.37i)15-s + (0.0434 + 0.246i)16-s + (−0.318 + 0.267i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 - 0.787i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.615 - 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.615 - 0.787i$
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: \((2,\ 74,\ (\ :3/2),\ 0.615 - 0.787i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.28228 + 0.625308i\)
\(L(\frac12)\) \(\approx\) \(1.28228 + 0.625308i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-222. + 33.9i)T \)
good3 \( 1 + (-6.42 + 2.33i)T + (20.6 - 17.3i)T^{2} \)
5 \( 1 + (3.55 - 20.1i)T + (-117. - 42.7i)T^{2} \)
7 \( 1 + (4.31 - 24.4i)T + (-322. - 117. i)T^{2} \)
11 \( 1 + (11.5 + 19.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-11.6 - 9.73i)T + (381. + 2.16e3i)T^{2} \)
17 \( 1 + (22.3 - 18.7i)T + (853. - 4.83e3i)T^{2} \)
19 \( 1 + (-145. + 52.8i)T + (5.25e3 - 4.40e3i)T^{2} \)
23 \( 1 + (-91.8 + 159. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (24.9 + 43.2i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 171.T + 2.97e4T^{2} \)
41 \( 1 + (50.7 + 42.5i)T + (1.19e4 + 6.78e4i)T^{2} \)
43 \( 1 + 22.3T + 7.95e4T^{2} \)
47 \( 1 + (242. - 419. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (0.712 + 4.04i)T + (-1.39e5 + 5.09e4i)T^{2} \)
59 \( 1 + (-45.6 - 258. i)T + (-1.92e5 + 7.02e4i)T^{2} \)
61 \( 1 + (-137. - 115. i)T + (3.94e4 + 2.23e5i)T^{2} \)
67 \( 1 + (-65.9 + 373. i)T + (-2.82e5 - 1.02e5i)T^{2} \)
71 \( 1 + (896. - 326. i)T + (2.74e5 - 2.30e5i)T^{2} \)
73 \( 1 - 733.T + 3.89e5T^{2} \)
79 \( 1 + (-93.2 + 529. i)T + (-4.63e5 - 1.68e5i)T^{2} \)
83 \( 1 + (-574. + 482. i)T + (9.92e4 - 5.63e5i)T^{2} \)
89 \( 1 + (124. + 707. i)T + (-6.62e5 + 2.41e5i)T^{2} \)
97 \( 1 + (189. - 328. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−14.38381022835992298128341200487, −13.30202682663045407584949839728, −11.80164277343442075421794089670, −10.89085029482906976240972225026, −9.519391419409266189235483122288, −8.483195384174788141688304799681, −7.46276801987943370007777919097, −6.31048993883379948895049611116, −3.05815456077191075036462489228, −2.58662138128592372200852831257, 1.09036366177258325475959864173, 3.65327608080925891778463192900, 5.05449550937566126913814645713, 7.46847098092942882726440439640, 8.202189055564018610509109217148, 9.357627374311410879991123154983, 9.907201496960131574379346907065, 11.67361242573472457646473153664, 13.17475555657980771359390570781, 13.80963681611100603514347685812

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