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Properties

Label 2-3675-735.188-c0-0-1
Degree $2$
Conductor $3675$
Sign $-0.103 + 0.994i$
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  + (0.330 − 0.943i)3-s + (0.974 + 0.222i)4-s + (0.111 − 0.993i)7-s + (−0.781 − 0.623i)9-s + (0.532 − 0.846i)12-s + (−1.79 − 0.201i)13-s + (0.900 + 0.433i)16-s + 1.56·19-s + (−0.900 − 0.433i)21-s + (−0.846 + 0.532i)27-s + (0.330 − 0.943i)28-s − 1.94i·31-s + (−0.623 − 0.781i)36-s + (1.65 + 1.03i)37-s + (−0.781 + 1.62i)39-s + ⋯
L(s)  = 1  + (0.330 − 0.943i)3-s + (0.974 + 0.222i)4-s + (0.111 − 0.993i)7-s + (−0.781 − 0.623i)9-s + (0.532 − 0.846i)12-s + (−1.79 − 0.201i)13-s + (0.900 + 0.433i)16-s + 1.56·19-s + (−0.900 − 0.433i)21-s + (−0.846 + 0.532i)27-s + (0.330 − 0.943i)28-s − 1.94i·31-s + (−0.623 − 0.781i)36-s + (1.65 + 1.03i)37-s + (−0.781 + 1.62i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.103 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.103 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-0.103 + 0.994i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (2393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ -0.103 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.675839863\)
\(L(\frac12)\) \(\approx\) \(1.675839863\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.330 + 0.943i)T \)
5 \( 1 \)
7 \( 1 + (-0.111 + 0.993i)T \)
good2 \( 1 + (-0.974 - 0.222i)T^{2} \)
11 \( 1 + (0.222 - 0.974i)T^{2} \)
13 \( 1 + (1.79 + 0.201i)T + (0.974 + 0.222i)T^{2} \)
17 \( 1 + (0.433 + 0.900i)T^{2} \)
19 \( 1 - 1.56T + T^{2} \)
23 \( 1 + (-0.433 + 0.900i)T^{2} \)
29 \( 1 + (-0.900 + 0.433i)T^{2} \)
31 \( 1 + 1.94iT - T^{2} \)
37 \( 1 + (-1.65 - 1.03i)T + (0.433 + 0.900i)T^{2} \)
41 \( 1 + (0.623 - 0.781i)T^{2} \)
43 \( 1 + (0.516 + 1.47i)T + (-0.781 + 0.623i)T^{2} \)
47 \( 1 + (0.974 + 0.222i)T^{2} \)
53 \( 1 + (-0.433 + 0.900i)T^{2} \)
59 \( 1 + (-0.623 - 0.781i)T^{2} \)
61 \( 1 + (1.90 - 0.433i)T + (0.900 - 0.433i)T^{2} \)
67 \( 1 + (-0.613 + 0.613i)T - iT^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (0.0498 + 0.442i)T + (-0.974 + 0.222i)T^{2} \)
79 \( 1 + 0.445iT - T^{2} \)
83 \( 1 + (0.974 - 0.222i)T^{2} \)
89 \( 1 + (0.222 + 0.974i)T^{2} \)
97 \( 1 + (-0.881 - 0.881i)T + iT^{2} \)
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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

−7.988709900103148018456766992301, −7.67249429230015591551619984836, −7.25780113943123013763647253000, −6.52599114405036087174674701664, −5.72138946178594648825424744022, −4.74200539335158705509384475162, −3.58268280075959322469874399742, −2.79191776607325728349254849634, −2.05422762839599829984543957842, −0.891847886286178377393660181069, 1.73154241541898984188557988436, 2.82987461242995732927354654146, 3.00604205283568125322919553393, 4.47990977901571551288877123629, 5.21579805214443009045682585620, 5.68792176238715867114334156996, 6.71441299221663762663904046763, 7.54039983731989372803429547719, 8.089395457519736600423058041047, 9.150141819514603661911495857351

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