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Properties

Label 2-74-37.27-c3-0-7
Degree $2$
Conductor $74$
Sign $-0.969 + 0.245i$
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 + (1.55 + 2.69i)3-s + (1.99 + 3.46i)4-s + (−12.2 + 7.06i)5-s − 6.22i·6-s + (−13.5 − 23.3i)7-s − 7.99i·8-s + (8.66 − 15.0i)9-s + 28.2·10-s − 40.4·11-s + (−6.22 + 10.7i)12-s + (−12.6 + 7.27i)13-s + 54.0i·14-s + (−38.0 − 21.9i)15-s + (−8 + 13.8i)16-s + (−56.7 − 32.7i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.299 + 0.518i)3-s + (0.249 + 0.433i)4-s + (−1.09 + 0.632i)5-s − 0.423i·6-s + (−0.729 − 1.26i)7-s − 0.353i·8-s + (0.320 − 0.555i)9-s + 0.893·10-s − 1.10·11-s + (−0.149 + 0.259i)12-s + (−0.268 + 0.155i)13-s + 1.03i·14-s + (−0.655 − 0.378i)15-s + (−0.125 + 0.216i)16-s + (−0.809 − 0.467i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.245i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.969 + 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.969 + 0.245i$
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: \((2,\ 74,\ (\ :3/2),\ -0.969 + 0.245i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0159441 - 0.128104i\)
\(L(\frac12)\) \(\approx\) \(0.0159441 - 0.128104i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-108. + 196. i)T \)
good3 \( 1 + (-1.55 - 2.69i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (12.2 - 7.06i)T + (62.5 - 108. i)T^{2} \)
7 \( 1 + (13.5 + 23.3i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + 40.4T + 1.33e3T^{2} \)
13 \( 1 + (12.6 - 7.27i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (56.7 + 32.7i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (39.7 - 22.9i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 - 57.6iT - 1.21e4T^{2} \)
29 \( 1 - 224. iT - 2.43e4T^{2} \)
31 \( 1 + 89.6iT - 2.97e4T^{2} \)
41 \( 1 + (36.5 + 63.2i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + 333. iT - 7.95e4T^{2} \)
47 \( 1 + 291.T + 1.03e5T^{2} \)
53 \( 1 + (357. - 618. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-653. - 377. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (549. - 317. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (152. + 263. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (531. + 919. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 942.T + 3.89e5T^{2} \)
79 \( 1 + (412. - 237. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-463. + 803. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-88.5 - 51.1i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 139. iT - 9.12e5T^{2} \)
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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

−13.48920781091970555774288427263, −12.34370991398315382047537938193, −10.94625748990001881027559197867, −10.34285301296514279339504673435, −9.176797218811973923198366023820, −7.65076763575292057476182331429, −6.90041027863080176910206199206, −4.19210724337642253624011791110, −3.17522463814427315019669132336, −0.091920851417949663977276252542, 2.47196180874764368578201592411, 4.85883662707777880342246074257, 6.49746210831021126434417808294, 8.005633109807197703096743680729, 8.437612775530052069527874764964, 9.866377030571792682035612377259, 11.31403556733733931981234606615, 12.56927145999906733126180828105, 13.16842765070273643732887065112, 15.00464081635019662328289788382

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