Properties

Label 2-3920-140.83-c0-0-2
Degree $2$
Conductor $3920$
Sign $0.704 + 0.709i$
Analytic cond. $1.95633$
Root an. cond. $1.39869$
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  + (0.923 + 0.382i)5-s i·9-s + (0.541 − 0.541i)13-s + (−1.30 − 1.30i)17-s + (0.707 + 0.707i)25-s − 1.41i·29-s + (1.41 − 1.41i)37-s + 0.765i·41-s + (0.382 − 0.923i)45-s + (−1 − i)53-s + 1.84i·61-s + (0.707 − 0.292i)65-s + (−1.30 + 1.30i)73-s − 81-s + (−0.707 − 1.70i)85-s + ⋯
L(s)  = 1  + (0.923 + 0.382i)5-s i·9-s + (0.541 − 0.541i)13-s + (−1.30 − 1.30i)17-s + (0.707 + 0.707i)25-s − 1.41i·29-s + (1.41 − 1.41i)37-s + 0.765i·41-s + (0.382 − 0.923i)45-s + (−1 − i)53-s + 1.84i·61-s + (0.707 − 0.292i)65-s + (−1.30 + 1.30i)73-s − 81-s + (−0.707 − 1.70i)85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $0.704 + 0.709i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (783, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ 0.704 + 0.709i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.483887867\)
\(L(\frac12)\) \(\approx\) \(1.483887867\)
\(L(1)\) 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 \( 1 + (-0.923 - 0.382i)T \)
7 \( 1 \)
good3 \( 1 + iT^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
17 \( 1 + (1.30 + 1.30i)T + iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
41 \( 1 - 0.765iT - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (1 + i)T + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.84iT - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - 1.84T + T^{2} \)
97 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
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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

−8.773838263378353436824161751871, −7.74238613772110780451369992948, −6.97146316509373695642585257973, −6.25346788586994715659972515086, −5.82389687482335868153518482127, −4.80117939744244548801327841661, −3.93458766718382285327602991611, −2.91223489637797268062145052274, −2.24712535752377705102801524397, −0.860143963808678002473206125729, 1.52966162116759110982921459493, 2.09412556922186869735400329816, 3.21721838160645157858781019448, 4.45486307954268323632521299033, 4.84218862036974696461527659015, 5.93230159192083491690770657785, 6.34361830506001743914063444148, 7.22467916138317068368960589988, 8.229478655216381607833906830019, 8.688611884882502080297065120248

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