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Properties

Label 2-3700-37.6-c0-0-2
Degree $2$
Conductor $3700$
Sign $-0.763 + 0.646i$
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  − i·3-s − 7-s − i·11-s + i·21-s − i·27-s + (1 − i)31-s − 33-s − 37-s − i·41-s + (−1 − i)43-s + 47-s − 53-s + (−1 + i)61-s − 71-s + i·73-s + ⋯
L(s)  = 1  − i·3-s − 7-s − i·11-s + i·21-s − i·27-s + (1 − i)31-s − 33-s − 37-s − i·41-s + (−1 − i)43-s + 47-s − 53-s + (−1 + i)61-s − 71-s + i·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.763 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.763 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3700\)    =    \(2^{2} \cdot 5^{2} \cdot 37\)
Sign: $-0.763 + 0.646i$
Analytic conductor: \(1.84654\)
Root analytic conductor: \(1.35887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3700} (1301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3700,\ (\ :0),\ -0.763 + 0.646i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9257873133\)
\(L(\frac12)\) \(\approx\) \(0.9257873133\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.462061145537324292456693315368, −7.47684334403087167650551057726, −7.03161890045280713856238587805, −6.15089744435409039550743599263, −5.82608902885731524797912321390, −4.57122643888033291710972523313, −3.57669975035337269975128605123, −2.81957916055466802801960885433, −1.76591776709781833021692025444, −0.52924771052633181650433947663, 1.59003838115123936655575137163, 2.93148261721247660301496297414, 3.53932253970631928763331513597, 4.56545167579348133333519823399, 4.90717122203873058364650217437, 6.06749756967490325645273229757, 6.74257073116249535442334034237, 7.42901460281978116693151751622, 8.405799624998082587348024356488, 9.211864993650701309140128221714

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