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Properties

Label 2-3675-735.179-c0-0-0
Degree $2$
Conductor $3675$
Sign $0.247 - 0.968i$
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  + (−0.294 + 0.955i)3-s + (0.988 − 0.149i)4-s + (−0.563 − 0.826i)7-s + (−0.826 − 0.563i)9-s + (−0.149 + 0.988i)12-s + (−0.636 + 1.32i)13-s + (0.955 − 0.294i)16-s + (0.826 + 1.43i)19-s + (0.955 − 0.294i)21-s + (0.781 − 0.623i)27-s + (−0.680 − 0.733i)28-s + (−0.623 + 1.07i)31-s + (−0.900 − 0.433i)36-s + (−0.108 + 0.722i)37-s + (−1.07 − 0.997i)39-s + ⋯
L(s)  = 1  + (−0.294 + 0.955i)3-s + (0.988 − 0.149i)4-s + (−0.563 − 0.826i)7-s + (−0.826 − 0.563i)9-s + (−0.149 + 0.988i)12-s + (−0.636 + 1.32i)13-s + (0.955 − 0.294i)16-s + (0.826 + 1.43i)19-s + (0.955 − 0.294i)21-s + (0.781 − 0.623i)27-s + (−0.680 − 0.733i)28-s + (−0.623 + 1.07i)31-s + (−0.900 − 0.433i)36-s + (−0.108 + 0.722i)37-s + (−1.07 − 0.997i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.247 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.247 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $0.247 - 0.968i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (1649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ 0.247 - 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.278437262\)
\(L(\frac12)\) \(\approx\) \(1.278437262\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.294 - 0.955i)T \)
5 \( 1 \)
7 \( 1 + (0.563 + 0.826i)T \)
good2 \( 1 + (-0.988 + 0.149i)T^{2} \)
11 \( 1 + (-0.365 + 0.930i)T^{2} \)
13 \( 1 + (0.636 - 1.32i)T + (-0.623 - 0.781i)T^{2} \)
17 \( 1 + (-0.733 + 0.680i)T^{2} \)
19 \( 1 + (-0.826 - 1.43i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.733 - 0.680i)T^{2} \)
29 \( 1 + (0.222 + 0.974i)T^{2} \)
31 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.108 - 0.722i)T + (-0.955 - 0.294i)T^{2} \)
41 \( 1 + (0.900 - 0.433i)T^{2} \)
43 \( 1 + (-1.75 - 0.400i)T + (0.900 + 0.433i)T^{2} \)
47 \( 1 + (-0.988 + 0.149i)T^{2} \)
53 \( 1 + (0.955 - 0.294i)T^{2} \)
59 \( 1 + (-0.0747 - 0.997i)T^{2} \)
61 \( 1 + (-1.95 - 0.294i)T + (0.955 + 0.294i)T^{2} \)
67 \( 1 + (-1.65 - 0.955i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.222 - 0.974i)T^{2} \)
73 \( 1 + (-0.728 - 0.0546i)T + (0.988 + 0.149i)T^{2} \)
79 \( 1 + (0.988 + 1.71i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.623 - 0.781i)T^{2} \)
89 \( 1 + (-0.365 - 0.930i)T^{2} \)
97 \( 1 + 0.149iT - T^{2} \)
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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

−9.087384907310793484642090310529, −8.052787966172690675497945858774, −7.18913917389443676685570724677, −6.68546588877754556699604749161, −5.89009835573894495213669179156, −5.15298509512268890923095440298, −4.12505781083742675092968281441, −3.54426216192000005604131012867, −2.59851797117659754940403188305, −1.33773511231653310813865059051, 0.78358928242562494509975653641, 2.39365258560328619792312200088, 2.54429602997223779732259017860, 3.59544277732145510202580839822, 5.30512841491737158803518153793, 5.54286494386336744687659166817, 6.43961211798443960071089877313, 7.09600207218949299437671956346, 7.64113686802942372120082222627, 8.317322726312303729658261553778

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