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Properties

Label 2-74-37.11-c1-0-1
Degree $2$
Conductor $74$
Sign $0.896 - 0.443i$
Analytic cond. $0.590892$
Root an. cond. $0.768695$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (−1.36 + 2.36i)3-s + (0.499 − 0.866i)4-s + (1.5 + 0.866i)5-s + 2.73i·6-s + (1 − 1.73i)7-s − 0.999i·8-s + (−2.23 − 3.86i)9-s + 1.73·10-s − 4.73·11-s + (1.36 + 2.36i)12-s + (−3 − 1.73i)13-s − 1.99i·14-s + (−4.09 + 2.36i)15-s + (−0.5 − 0.866i)16-s + (6.69 − 3.86i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.788 + 1.36i)3-s + (0.249 − 0.433i)4-s + (0.670 + 0.387i)5-s + 1.11i·6-s + (0.377 − 0.654i)7-s − 0.353i·8-s + (−0.744 − 1.28i)9-s + 0.547·10-s − 1.42·11-s + (0.394 + 0.683i)12-s + (−0.832 − 0.480i)13-s − 0.534i·14-s + (−1.05 + 0.610i)15-s + (−0.125 − 0.216i)16-s + (1.62 − 0.937i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.896 - 0.443i$
Analytic conductor: \(0.590892\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :1/2),\ 0.896 - 0.443i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04759 + 0.244929i\)
\(L(\frac12)\) \(\approx\) \(1.04759 + 0.244929i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
37 \( 1 + (5.69 + 2.13i)T \)
good3 \( 1 + (1.36 - 2.36i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1 + 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 4.73T + 11T^{2} \)
13 \( 1 + (3 + 1.73i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-6.69 + 3.86i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.09 - 0.633i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.73iT - 23T^{2} \)
29 \( 1 - 8.66iT - 29T^{2} \)
31 \( 1 + 1.26iT - 31T^{2} \)
41 \( 1 + (4.96 - 8.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 0.928iT - 43T^{2} \)
47 \( 1 - 4.73T + 47T^{2} \)
53 \( 1 + (1.26 + 2.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.19 + 1.26i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.5 - 0.866i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.09 + 8.83i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.73 - 3i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 + (11.4 + 6.63i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.83 - 4.90i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.89 - 3.40i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 7.73iT - 97T^{2} \)
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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

−14.62929056130012672685522010831, −13.75265949837372006420062638798, −12.33829517681194657827370703785, −11.13025562576389658608651839975, −10.22572874941855798934882086664, −9.834822439316199948385952778723, −7.48207459630864770676611344149, −5.52795291595256257150481002472, −4.99630177260622274496114811717, −3.21896739138208407882888504935, 2.17634053710953356201098501685, 5.24088143535986636179282696238, 5.85627710721084722349181152484, 7.28688928546112566849819576227, 8.287451503679754829241604697852, 10.22748427003330092383453510609, 11.79891918662957083885896220858, 12.45141581263461391815434389542, 13.23251518755366938539411013694, 14.22475780935800700136610272679

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