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Properties

Label 2-3700-37.6-c0-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(1.541707645\)
\(L(\frac12)\) \(\approx\) \(1.541707645\)
\(L(1)\) 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

−8.956584723532563913390016677727, −8.020182439039069591114751630904, −7.58825631936971350716482212973, −6.38903690162052707381245535998, −5.70032833238008057227242691455, −4.83012257246923250519507294244, −4.30954461684620370544057341372, −3.47894093358043199493904652734, −2.47157464524926324940072655438, −1.16837884808913097938400483206, 1.19996283865262811193114426908, 1.91664682300479078087942625926, 2.82970047771649441557787125453, 4.22886072405841428887416362721, 4.75095115786952992130065307046, 5.69313444720268379451392259079, 6.65132866747826909913809702942, 7.12228974518868198769369499945, 7.929813638148986413449549770875, 8.271323530141024550684178682238

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