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Properties

Label 2-74-37.26-c5-0-2
Degree $2$
Conductor $74$
Sign $-0.996 + 0.0803i$
Analytic cond. $11.8684$
Root an. cond. $3.44505$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 + 3.46i)2-s + (−1.88 + 3.25i)3-s + (−7.99 + 13.8i)4-s + (7.78 − 13.4i)5-s − 15.0·6-s + (−28.3 + 49.0i)7-s − 63.9·8-s + (114. + 198. i)9-s + 62.3·10-s − 674.·11-s + (−30.0 − 52.1i)12-s + (−195. + 339. i)13-s − 226.·14-s + (29.2 + 50.7i)15-s + (−128 − 221. i)16-s + (−745. − 1.29e3i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.120 + 0.208i)3-s + (−0.249 + 0.433i)4-s + (0.139 − 0.241i)5-s − 0.170·6-s + (−0.218 + 0.378i)7-s − 0.353·8-s + (0.470 + 0.815i)9-s + 0.197·10-s − 1.68·11-s + (−0.0603 − 0.104i)12-s + (−0.321 + 0.556i)13-s − 0.308·14-s + (0.0336 + 0.0582i)15-s + (−0.125 − 0.216i)16-s + (−0.625 − 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0803i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.996 + 0.0803i$
Analytic conductor: \(11.8684\)
Root analytic conductor: \(3.44505\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :5/2),\ -0.996 + 0.0803i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0385127 - 0.956598i\)
\(L(\frac12)\) \(\approx\) \(0.0385127 - 0.956598i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2 - 3.46i)T \)
37 \( 1 + (8.11e3 - 1.87e3i)T \)
good3 \( 1 + (1.88 - 3.25i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (-7.78 + 13.4i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (28.3 - 49.0i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + 674.T + 1.61e5T^{2} \)
13 \( 1 + (195. - 339. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (745. + 1.29e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (142. - 246. i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + 2.91e3T + 6.43e6T^{2} \)
29 \( 1 - 4.00e3T + 2.05e7T^{2} \)
31 \( 1 + 529.T + 2.86e7T^{2} \)
41 \( 1 + (8.64e3 - 1.49e4i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 - 1.25e4T + 1.47e8T^{2} \)
47 \( 1 - 1.57e4T + 2.29e8T^{2} \)
53 \( 1 + (-1.47e4 - 2.55e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-3.75e3 - 6.50e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-2.74e4 + 4.75e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (8.15e3 - 1.41e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (-1.81e4 + 3.15e4i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 - 5.64e4T + 2.07e9T^{2} \)
79 \( 1 + (4.50e4 - 7.80e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (4.08e4 + 7.07e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + (537. + 931. i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 1.22e5T + 8.58e9T^{2} \)
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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

−14.00044964093314802317557939552, −13.25217330294169210314424468198, −12.21416282137143785063833368403, −10.76890779510376176596079649918, −9.589540278190634198455522088512, −8.209938462071561826362971671148, −7.10844844084331498184432606751, −5.50279794438454802250137363848, −4.60762181751605119520930970724, −2.51839881955153376040334232167, 0.35551961202761135457435488469, 2.37493871628129438089291796734, 3.95799331362707744227960300292, 5.56803899583956734061970146114, 6.93298403048975176625301802732, 8.430900822841585986900801690325, 10.16392158575785708427065313451, 10.55241337432905146756238588446, 12.18846820949456233570152876018, 12.88818473603949821506859870434

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