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Properties

Label 2-3675-735.122-c0-0-1
Degree $2$
Conductor $3675$
Sign $0.781 + 0.624i$
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  + (0.399 − 0.916i)3-s + (0.563 + 0.826i)4-s + (0.999 − 0.0373i)7-s + (−0.680 − 0.733i)9-s + (0.982 − 0.185i)12-s + (1.67 − 1.05i)13-s + (−0.365 + 0.930i)16-s + (−0.680 − 1.17i)19-s + (0.365 − 0.930i)21-s + (−0.943 + 0.330i)27-s + (0.593 + 0.804i)28-s + (−0.751 − 0.433i)31-s + (0.222 − 0.974i)36-s + (0.370 + 1.95i)37-s + (−0.294 − 1.95i)39-s + ⋯
L(s)  = 1  + (0.399 − 0.916i)3-s + (0.563 + 0.826i)4-s + (0.999 − 0.0373i)7-s + (−0.680 − 0.733i)9-s + (0.982 − 0.185i)12-s + (1.67 − 1.05i)13-s + (−0.365 + 0.930i)16-s + (−0.680 − 1.17i)19-s + (0.365 − 0.930i)21-s + (−0.943 + 0.330i)27-s + (0.593 + 0.804i)28-s + (−0.751 − 0.433i)31-s + (0.222 − 0.974i)36-s + (0.370 + 1.95i)37-s + (−0.294 − 1.95i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $0.781 + 0.624i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (857, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ 0.781 + 0.624i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.924409435\)
\(L(\frac12)\) \(\approx\) \(1.924409435\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.399 + 0.916i)T \)
5 \( 1 \)
7 \( 1 + (-0.999 + 0.0373i)T \)
good2 \( 1 + (-0.563 - 0.826i)T^{2} \)
11 \( 1 + (-0.0747 - 0.997i)T^{2} \)
13 \( 1 + (-1.67 + 1.05i)T + (0.433 - 0.900i)T^{2} \)
17 \( 1 + (0.149 - 0.988i)T^{2} \)
19 \( 1 + (0.680 + 1.17i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.149 - 0.988i)T^{2} \)
29 \( 1 + (0.623 - 0.781i)T^{2} \)
31 \( 1 + (0.751 + 0.433i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.370 - 1.95i)T + (-0.930 + 0.365i)T^{2} \)
41 \( 1 + (-0.222 - 0.974i)T^{2} \)
43 \( 1 + (1.93 + 0.218i)T + (0.974 + 0.222i)T^{2} \)
47 \( 1 + (0.563 + 0.826i)T^{2} \)
53 \( 1 + (0.930 + 0.365i)T^{2} \)
59 \( 1 + (-0.955 + 0.294i)T^{2} \)
61 \( 1 + (0.634 - 0.930i)T + (-0.365 - 0.930i)T^{2} \)
67 \( 1 + (-1.79 + 0.481i)T + (0.866 - 0.5i)T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (0.0698 + 0.132i)T + (-0.563 + 0.826i)T^{2} \)
79 \( 1 + (1.43 - 0.826i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.433 + 0.900i)T^{2} \)
89 \( 1 + (-0.0747 + 0.997i)T^{2} \)
97 \( 1 + (-1.35 + 1.35i)T - iT^{2} \)
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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

−8.499622977982676873374094142671, −7.986719403878843685295173420873, −7.25986952405650618317258970427, −6.54250281120121429203050105318, −5.90222908362550623856200687014, −4.81877711947732562620236664653, −3.71725033196078871619739000912, −3.04845975669908596007714619904, −2.12086344573562026178761819309, −1.19771634821599413979445998547, 1.56750712821860879362055722533, 2.11280234827145969470582255942, 3.50617333796007746810358745771, 4.15274458125251469529217296258, 5.01958989528421579741352111606, 5.76123996731148741189207705726, 6.37235218496395686842339867392, 7.37909883088053737130322300321, 8.284212765245571040419531439404, 8.775694829128521365717785992935

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