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Properties

Label 2-3700-740.23-c0-0-0
Degree $2$
Conductor $3700$
Sign $0.00214 - 0.999i$
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 + (−0.366 − 0.366i)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 + (1.36 − 0.366i)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 + (−0.366 − 0.366i)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 + (1.36 − 0.366i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00214 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00214 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3700\)    =    \(2^{2} \cdot 5^{2} \cdot 37\)
Sign: $0.00214 - 0.999i$
Analytic conductor: \(1.84654\)
Root analytic conductor: \(1.35887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3700} (2243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3700,\ (\ :0),\ 0.00214 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.230175336\)
\(L(\frac12)\) \(\approx\) \(2.230175336\)
\(L(1)\) 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 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
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 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (1.86 + 0.5i)T + (0.866 + 0.5i)T^{2} \)
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 \( 1 + (0.5 - 0.133i)T + (0.866 - 0.5i)T^{2} \)
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

−8.699392344330429433238845241828, −7.913799029393253964183523644013, −7.34270092077696640200414109778, −6.62720171493375899684254850927, −5.88631361975778330899983581780, −5.11308147018239799802673424525, −4.28419056230916022995080388534, −3.76578334414380738175481936607, −2.58480761748897134067001321638, −1.71114282316159398577028843638, 1.01569777304532563367244706446, 2.12913942530925576557983242255, 3.04606007709590653142412796751, 3.97613326194533055134601868448, 4.54423795827166947729060764740, 5.42308971380183987536545571423, 6.16630630524096794131161258355, 7.04973311115679630600332227704, 7.39122708870693502589553379190, 8.710634831505934995312334232435

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