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Properties

Label 2-3700-740.347-c0-0-0
Degree $2$
Conductor $3700$
Sign $0.292 + 0.956i$
Analytic cond. $1.84654$
Root an. cond. $1.35887$
Motivic weight $0$
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 + (0.499 + 0.866i)4-s − 0.999i·8-s + (−0.866 − 0.5i)9-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.499 + 0.866i)18-s + (1.36 − 1.36i)29-s + (0.866 − 0.499i)32-s + (0.866 − 0.499i)34-s − 0.999i·36-s + (0.866 − 0.5i)37-s + (1.5 − 0.866i)41-s + (−0.866 − 0.5i)49-s + (−0.366 − 1.36i)53-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s − 0.999i·8-s + (−0.866 − 0.5i)9-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.499 + 0.866i)18-s + (1.36 − 1.36i)29-s + (0.866 − 0.499i)32-s + (0.866 − 0.499i)34-s − 0.999i·36-s + (0.866 − 0.5i)37-s + (1.5 − 0.866i)41-s + (−0.866 − 0.5i)49-s + (−0.366 − 1.36i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3700\)    =    \(2^{2} \cdot 5^{2} \cdot 37\)
Sign: $0.292 + 0.956i$
Analytic conductor: \(1.84654\)
Root analytic conductor: \(1.35887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3700} (3307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3700,\ (\ :0),\ 0.292 + 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7066067661\)
\(L(\frac12)\) \(\approx\) \(0.7066067661\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 \)
37 \( 1 + (-0.866 + 0.5i)T \)
good3 \( 1 + (0.866 + 0.5i)T^{2} \)
7 \( 1 + (0.866 + 0.5i)T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
31 \( 1 - iT^{2} \)
41 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (0.133 - 0.5i)T + (-0.866 - 0.5i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-1 - i)T + iT^{2} \)
79 \( 1 + (0.866 + 0.5i)T^{2} \)
83 \( 1 + (0.866 - 0.5i)T^{2} \)
89 \( 1 + (0.5 + 1.86i)T + (-0.866 + 0.5i)T^{2} \)
97 \( 1 - 1.73T + T^{2} \)
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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

−8.490548370178034188800370655686, −8.158829901910220661067472070608, −7.22291948598619160063246340474, −6.36555869723118341921555104845, −5.85914437662118805665244560444, −4.51213308020335398300577713798, −3.73479125326380864409426117049, −2.81413482607366074624121339362, −2.03084534267029494748203269663, −0.63715488049539210376889932840, 1.06564558925585274278139865091, 2.38937685184932740618524699464, 3.04665899878209660345667356242, 4.63403413753255771279975077233, 5.14000427105691417025556327076, 6.14471679369044540232954315180, 6.60623868799468579497116276098, 7.63009554727343376461733673327, 8.015148821316890787300850526453, 8.933133648121211777745864972127

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