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Properties

Label 2-3680-115.114-c0-0-1
Degree $2$
Conductor $3680$
Sign $-0.707 - 0.707i$
Analytic cond. $1.83655$
Root an. cond. $1.35519$
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  + 1.41i·3-s + i·5-s + 1.41·7-s − 1.00·9-s − 1.41·15-s + 2.00i·21-s + (0.707 + 0.707i)23-s − 25-s − 2·29-s + 1.41i·35-s + 1.41·43-s − 1.00i·45-s − 1.41i·47-s + 1.00·49-s + 2i·61-s + ⋯
L(s)  = 1  + 1.41i·3-s + i·5-s + 1.41·7-s − 1.00·9-s − 1.41·15-s + 2.00i·21-s + (0.707 + 0.707i)23-s − 25-s − 2·29-s + 1.41i·35-s + 1.41·43-s − 1.00i·45-s − 1.41i·47-s + 1.00·49-s + 2i·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3680\)    =    \(2^{5} \cdot 5 \cdot 23\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3680} (2529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3680,\ (\ :0),\ -0.707 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.465288079\)
\(L(\frac12)\) \(\approx\) \(1.465288079\)
\(L(1)\) 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 \( 1 - iT \)
23 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 - 1.41iT - T^{2} \)
7 \( 1 - 1.41T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
29 \( 1 + 2T + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - 1.41T + T^{2} \)
47 \( 1 + 1.41iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - 2iT - T^{2} \)
67 \( 1 - 1.41T + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.41T + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.185011656366819057004722651379, −8.289766418167120323441580462607, −7.53719677485393465497867526051, −6.91182140120898536499860149278, −5.56568688111097945399740676029, −5.34159857376235789267827312960, −4.22632038965429708747826021741, −3.79649892183992764609763518039, −2.78429441309498527433453743490, −1.72864892486324085668323998487, 0.898793013711947164044818998452, 1.69201549300651204660739007246, 2.39194734127402469531436054638, 3.89943238926798088473593348898, 4.79404405812162345721164250856, 5.42579771580477089035382576214, 6.23100602016396711131122090659, 7.14783947168682036072420014203, 7.87109548028653309765548082995, 8.102426995493974857826306096839

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