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Properties

Label 2-3675-735.389-c0-0-0
Degree $2$
Conductor $3675$
Sign $0.329 - 0.944i$
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  + (0.997 + 0.0747i)3-s + (0.733 + 0.680i)4-s + (0.149 + 0.988i)7-s + (0.988 + 0.149i)9-s + (0.680 + 0.733i)12-s + (−1.29 + 1.03i)13-s + (0.0747 + 0.997i)16-s + (−0.988 − 1.71i)19-s + (0.0747 + 0.997i)21-s + (0.974 + 0.222i)27-s + (−0.563 + 0.826i)28-s + (0.222 − 0.385i)31-s + (0.623 + 0.781i)36-s + (1.29 + 1.40i)37-s + (−1.36 + 0.930i)39-s + ⋯
L(s)  = 1  + (0.997 + 0.0747i)3-s + (0.733 + 0.680i)4-s + (0.149 + 0.988i)7-s + (0.988 + 0.149i)9-s + (0.680 + 0.733i)12-s + (−1.29 + 1.03i)13-s + (0.0747 + 0.997i)16-s + (−0.988 − 1.71i)19-s + (0.0747 + 0.997i)21-s + (0.974 + 0.222i)27-s + (−0.563 + 0.826i)28-s + (0.222 − 0.385i)31-s + (0.623 + 0.781i)36-s + (1.29 + 1.40i)37-s + (−1.36 + 0.930i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $0.329 - 0.944i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (1124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ 0.329 - 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.107874232\)
\(L(\frac12)\) \(\approx\) \(2.107874232\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.997 - 0.0747i)T \)
5 \( 1 \)
7 \( 1 + (-0.149 - 0.988i)T \)
good2 \( 1 + (-0.733 - 0.680i)T^{2} \)
11 \( 1 + (-0.955 + 0.294i)T^{2} \)
13 \( 1 + (1.29 - 1.03i)T + (0.222 - 0.974i)T^{2} \)
17 \( 1 + (0.826 + 0.563i)T^{2} \)
19 \( 1 + (0.988 + 1.71i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.826 - 0.563i)T^{2} \)
29 \( 1 + (0.900 - 0.433i)T^{2} \)
31 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-1.29 - 1.40i)T + (-0.0747 + 0.997i)T^{2} \)
41 \( 1 + (-0.623 + 0.781i)T^{2} \)
43 \( 1 + (-0.541 + 1.12i)T + (-0.623 - 0.781i)T^{2} \)
47 \( 1 + (-0.733 - 0.680i)T^{2} \)
53 \( 1 + (0.0747 + 0.997i)T^{2} \)
59 \( 1 + (-0.365 + 0.930i)T^{2} \)
61 \( 1 + (-1.07 + 0.997i)T + (0.0747 - 0.997i)T^{2} \)
67 \( 1 + (-0.129 - 0.0747i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (1.77 - 0.698i)T + (0.733 - 0.680i)T^{2} \)
79 \( 1 + (0.733 + 1.26i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.222 - 0.974i)T^{2} \)
89 \( 1 + (-0.955 - 0.294i)T^{2} \)
97 \( 1 + 0.730iT - T^{2} \)
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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

−8.718905645850732619766907004723, −8.273667683786582312869615856249, −7.30799989280692192552803662577, −6.92564712032550437919842433755, −6.08230264218135197710135852198, −4.76698572888470541422906780704, −4.30465316224833022878234695831, −3.05713864427820450180550450318, −2.46547817357438439377740654395, −1.94057935817216066134634701802, 1.07551611303513596067166678743, 2.11203457635563034001877067069, 2.86126533484179673789037998881, 3.87508337459024503329528097408, 4.63592058076986411459634611944, 5.66091970039470883005128205820, 6.44149755040473706864098058538, 7.38383432721754551916488223912, 7.64344967118458373586048472433, 8.378029322272252495578305074362

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