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Properties

Label 2-3675-105.47-c0-0-8
Degree $2$
Conductor $3675$
Sign $0.358 + 0.933i$
Analytic cond. $1.83406$
Root an. cond. $1.35427$
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  + (−0.198 − 0.739i)2-s + (0.965 + 0.258i)3-s + (0.358 − 0.207i)4-s − 0.765i·6-s + (−0.765 − 0.765i)8-s + (0.866 + 0.499i)9-s + (0.400 − 0.107i)12-s + (−0.207 + 0.358i)16-s + (0.366 − 1.36i)17-s + (0.198 − 0.739i)18-s + (−0.923 + 1.60i)19-s + (1.78 − 0.478i)23-s + (−0.541 − 0.937i)24-s + (0.707 + 0.707i)27-s + (0.662 − 0.382i)31-s + (−0.739 − 0.198i)32-s + ⋯
L(s)  = 1  + (−0.198 − 0.739i)2-s + (0.965 + 0.258i)3-s + (0.358 − 0.207i)4-s − 0.765i·6-s + (−0.765 − 0.765i)8-s + (0.866 + 0.499i)9-s + (0.400 − 0.107i)12-s + (−0.207 + 0.358i)16-s + (0.366 − 1.36i)17-s + (0.198 − 0.739i)18-s + (−0.923 + 1.60i)19-s + (1.78 − 0.478i)23-s + (−0.541 − 0.937i)24-s + (0.707 + 0.707i)27-s + (0.662 − 0.382i)31-s + (−0.739 − 0.198i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $0.358 + 0.933i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (1832, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ 0.358 + 0.933i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.892490550\)
\(L(\frac12)\) \(\approx\) \(1.892490550\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.198 + 0.739i)T + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
19 \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-1.78 + 0.478i)T + (0.866 - 0.5i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-0.662 + 0.382i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.866 - 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
53 \( 1 + (-0.478 + 1.78i)T + (-0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.662 - 0.382i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.866 - 0.5i)T^{2} \)
79 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (1 - i)T - iT^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
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

−8.690725995894774375069252264208, −8.018359002800957266229355331207, −7.11498839339696071039355839599, −6.55448874447881140282809030084, −5.46488015874240291625253145814, −4.56489357943662815153869672736, −3.58852123864088898118608571124, −2.92434344954663343249354983655, −2.17966537098573523184142350656, −1.16530306750135656113068198507, 1.46489094962983220048873328407, 2.58940000979422647482356461787, 3.16463966273546431384793621663, 4.21499931207633856585370904113, 5.17192538405599595502023349438, 6.23026640449088163390289141624, 6.81762733414768713957109622937, 7.34125352245468315648252021273, 8.141993898596858639784401785843, 8.711166806186270867838696573526

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