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Properties

Label 2-3675-735.458-c0-0-1
Degree $2$
Conductor $3675$
Sign $0.854 - 0.519i$
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  + (0.884 − 0.467i)3-s + (0.149 + 0.988i)4-s + (0.982 + 0.185i)7-s + (0.563 − 0.826i)9-s + (0.593 + 0.804i)12-s + (−1.38 + 0.484i)13-s + (−0.955 + 0.294i)16-s + (0.563 + 0.975i)19-s + (0.955 − 0.294i)21-s + (0.111 − 0.993i)27-s + (−0.0373 + 0.999i)28-s + (1.35 + 0.781i)31-s + (0.900 + 0.433i)36-s + (1.49 − 1.10i)37-s + (−0.997 + 1.07i)39-s + ⋯
L(s)  = 1  + (0.884 − 0.467i)3-s + (0.149 + 0.988i)4-s + (0.982 + 0.185i)7-s + (0.563 − 0.826i)9-s + (0.593 + 0.804i)12-s + (−1.38 + 0.484i)13-s + (−0.955 + 0.294i)16-s + (0.563 + 0.975i)19-s + (0.955 − 0.294i)21-s + (0.111 − 0.993i)27-s + (−0.0373 + 0.999i)28-s + (1.35 + 0.781i)31-s + (0.900 + 0.433i)36-s + (1.49 − 1.10i)37-s + (−0.997 + 1.07i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 - 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 - 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $0.854 - 0.519i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (1193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ 0.854 - 0.519i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.963916755\)
\(L(\frac12)\) \(\approx\) \(1.963916755\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.884 + 0.467i)T \)
5 \( 1 \)
7 \( 1 + (-0.982 - 0.185i)T \)
good2 \( 1 + (-0.149 - 0.988i)T^{2} \)
11 \( 1 + (-0.365 + 0.930i)T^{2} \)
13 \( 1 + (1.38 - 0.484i)T + (0.781 - 0.623i)T^{2} \)
17 \( 1 + (0.680 + 0.733i)T^{2} \)
19 \( 1 + (-0.563 - 0.975i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.680 + 0.733i)T^{2} \)
29 \( 1 + (-0.222 - 0.974i)T^{2} \)
31 \( 1 + (-1.35 - 0.781i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-1.49 + 1.10i)T + (0.294 - 0.955i)T^{2} \)
41 \( 1 + (-0.900 + 0.433i)T^{2} \)
43 \( 1 + (0.734 - 0.461i)T + (0.433 - 0.900i)T^{2} \)
47 \( 1 + (0.149 + 0.988i)T^{2} \)
53 \( 1 + (-0.294 - 0.955i)T^{2} \)
59 \( 1 + (-0.0747 - 0.997i)T^{2} \)
61 \( 1 + (0.0444 - 0.294i)T + (-0.955 - 0.294i)T^{2} \)
67 \( 1 + (0.152 + 0.569i)T + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + (0.222 - 0.974i)T^{2} \)
73 \( 1 + (0.476 + 0.553i)T + (-0.149 + 0.988i)T^{2} \)
79 \( 1 + (1.71 - 0.988i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.781 + 0.623i)T^{2} \)
89 \( 1 + (-0.365 - 0.930i)T^{2} \)
97 \( 1 + (0.105 + 0.105i)T + iT^{2} \)
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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

−8.453937498393266408588525572547, −8.100791929758386652810247091697, −7.40374348537054378488660555535, −6.96051904139485642016967048531, −5.88564561358689388586277932211, −4.69785211444322721923968685214, −4.16560141550273518025596841110, −3.07618364629702112610191974912, −2.43301437805298664172000325085, −1.53723240512789001346432450721, 1.12941104201757571304621734514, 2.30352087319640480287763990674, 2.85153491986329118496766790637, 4.28283262246818524344493368907, 4.83364869188022000870175626509, 5.34599779865592292103676414868, 6.48228358303495191669731829506, 7.38937501649925501112296780719, 7.86323449793975793883123604966, 8.714288618903258517663848954395

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