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Properties

Label 2-3675-105.17-c0-0-9
Degree $2$
Conductor $3675$
Sign $-0.937 - 0.347i$
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  + (−0.965 + 0.258i)3-s + (−0.866 − 0.5i)4-s + (0.866 − 0.499i)9-s + (0.965 + 0.258i)12-s + (−1.41 − 1.41i)13-s + (0.499 + 0.866i)16-s + (−0.707 + 0.707i)27-s − 36-s + (1.73 + 1.00i)39-s + (−0.707 − 0.707i)48-s + (0.517 + 1.93i)52-s − 0.999i·64-s + (−1.93 + 0.517i)73-s + (−1.73 + i)79-s + (0.500 − 0.866i)81-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)3-s + (−0.866 − 0.5i)4-s + (0.866 − 0.499i)9-s + (0.965 + 0.258i)12-s + (−1.41 − 1.41i)13-s + (0.499 + 0.866i)16-s + (−0.707 + 0.707i)27-s − 36-s + (1.73 + 1.00i)39-s + (−0.707 − 0.707i)48-s + (0.517 + 1.93i)52-s − 0.999i·64-s + (−1.93 + 0.517i)73-s + (−1.73 + i)79-s + (0.500 − 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-0.937 - 0.347i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (2432, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ -0.937 - 0.347i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02738745698\)
\(L(\frac12)\) \(\approx\) \(0.02738745698\)
\(L(1)\) 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 \( 1 + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
17 \( 1 + (0.866 - 0.5i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.866 - 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (1.93 - 0.517i)T + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
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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

−8.300645647188718896893254812644, −7.54033081346563687367124185205, −6.72884395752522861285561020501, −5.73925612362870307369622447383, −5.34331111011240873690581310431, −4.66728676900791271386362798432, −3.89668809176168591840605309563, −2.73964056372123506810849260300, −1.23573260777014231187546855513, −0.02036363190131954997820051477, 1.60225585707809007510935911521, 2.76647261863151861012673908722, 4.07943212037464337815616061414, 4.57682152726398944800370395810, 5.24325718098362532151070372008, 6.11843267154454391934854449940, 7.09309967165045871584494586150, 7.40450858435501896700295332735, 8.376990200834800361428211260159, 9.194713343401768614727400463247

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