Properties

Label 2-3700-37.6-c0-0-1
Degree $2$
Conductor $3700$
Sign $0.646 + 0.763i$
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 + (1 + i)17-s + (1 + i)19-s i·21-s + (1 + i)23-s i·27-s + (−1 + i)29-s − 33-s i·37-s + i·41-s − 47-s + (1 − i)51-s − 53-s + ⋯
L(s)  = 1  i·3-s + 7-s i·11-s + (1 + i)17-s + (1 + i)19-s i·21-s + (1 + i)23-s i·27-s + (−1 + i)29-s − 33-s i·37-s + i·41-s − 47-s + (1 − i)51-s − 53-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3700\)    =    \(2^{2} \cdot 5^{2} \cdot 37\)
Sign: $0.646 + 0.763i$
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: \((2,\ 3700,\ (\ :0),\ 0.646 + 0.763i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.609501215\)
\(L(\frac12)\) \(\approx\) \(1.609501215\)
\(L(1)\) 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 \( 1 + iT \)
good3 \( 1 + iT - T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 + iT - T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (-1 - i)T + iT^{2} \)
19 \( 1 + (-1 - i)T + iT^{2} \)
23 \( 1 + (-1 - i)T + iT^{2} \)
29 \( 1 + (1 - i)T - iT^{2} \)
31 \( 1 - iT^{2} \)
41 \( 1 - iT - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + T + T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 - iT^{2} \)
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 \( 1 + iT^{2} \)
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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

−8.278530297918887055281650979792, −7.82784466679219195638833579332, −7.36352551910404613276500251614, −6.37647011753488254974639209808, −5.64531303540702486790841932689, −5.05966832635472737334370281371, −3.79079293939286624844046370711, −3.11994087108081785672116673651, −1.61827613678575879877872910703, −1.31929421655682721656076924638, 1.25082275505334910550519481298, 2.48567393512262704333019007315, 3.43369668097751419319834170452, 4.44911323524444647214184036062, 4.93846139880544497394701628179, 5.39425221866347838874920490589, 6.76955636791342951008482835169, 7.36497555774530328737722817188, 8.042577099153215325023264020322, 8.986853217855073298170146806100

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