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Properties

Label 2-74-37.27-c3-0-2
Degree $2$
Conductor $74$
Sign $-0.311 - 0.950i$
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  + (1.73 + i)2-s + (2.94 + 5.10i)3-s + (1.99 + 3.46i)4-s + (−12.2 + 7.06i)5-s + 11.7i·6-s + (0.0476 + 0.0825i)7-s + 7.99i·8-s + (−3.85 + 6.67i)9-s − 28.2·10-s + 1.95·11-s + (−11.7 + 20.4i)12-s + (48.4 − 27.9i)13-s + 0.190i·14-s + (−72.0 − 41.6i)15-s + (−8 + 13.8i)16-s + (94.2 + 54.4i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.566 + 0.981i)3-s + (0.249 + 0.433i)4-s + (−1.09 + 0.631i)5-s + 0.801i·6-s + (0.00257 + 0.00445i)7-s + 0.353i·8-s + (−0.142 + 0.247i)9-s − 0.893·10-s + 0.0536·11-s + (−0.283 + 0.490i)12-s + (1.03 − 0.596i)13-s + 0.00364i·14-s + (−1.24 − 0.716i)15-s + (−0.125 + 0.216i)16-s + (1.34 + 0.776i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.311 - 0.950i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.311 - 0.950i$
Analytic conductor: \(4.36614\)
Root analytic conductor: \(2.08953\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :3/2),\ -0.311 - 0.950i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.22869 + 1.69564i\)
\(L(\frac12)\) \(\approx\) \(1.22869 + 1.69564i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.73 - i)T \)
37 \( 1 + (203. + 95.1i)T \)
good3 \( 1 + (-2.94 - 5.10i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (12.2 - 7.06i)T + (62.5 - 108. i)T^{2} \)
7 \( 1 + (-0.0476 - 0.0825i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 - 1.95T + 1.33e3T^{2} \)
13 \( 1 + (-48.4 + 27.9i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-94.2 - 54.4i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (16.5 - 9.56i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 - 112. iT - 1.21e4T^{2} \)
29 \( 1 + 166. iT - 2.43e4T^{2} \)
31 \( 1 + 237. iT - 2.97e4T^{2} \)
41 \( 1 + (79.1 + 137. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + 72.7iT - 7.95e4T^{2} \)
47 \( 1 + 0.209T + 1.03e5T^{2} \)
53 \( 1 + (62.6 - 108. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (345. + 199. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (421. - 243. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-454. - 787. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-53.5 - 92.7i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 653.T + 3.89e5T^{2} \)
79 \( 1 + (-735. + 424. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (280. - 485. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-109. - 63.1i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.49e3iT - 9.12e5T^{2} \)
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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

−14.76330351742766736716129410708, −13.59497297866249274462772108435, −12.20878004467624569356862863703, −11.10833398901588776224508408787, −10.01457730069533416628414067822, −8.462529974555560472512959603644, −7.49565713578051821952414922045, −5.81833258127989196244695055143, −3.97993970964241147542620904479, −3.39340138367132498543465118060, 1.26391114704714747585593494893, 3.28269671146856482657001847968, 4.80562443364898933009704266193, 6.69590863712106306128834897249, 7.88255286669791877371007604270, 8.873354367897079138806177137031, 10.72191847660374329855723007842, 12.04237450418752929543372491234, 12.51856410613421130656089246117, 13.69840629230114552949989521629

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