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Properties

Label 2-3675-735.143-c0-0-1
Degree $2$
Conductor $3675$
Sign $0.599 + 0.800i$
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  + (−0.999 + 0.0373i)3-s + (0.930 − 0.365i)4-s + (0.757 − 0.652i)7-s + (0.997 − 0.0747i)9-s + (−0.916 + 0.399i)12-s + (0.631 − 1.80i)13-s + (0.733 − 0.680i)16-s + (0.997 + 1.72i)19-s + (−0.733 + 0.680i)21-s + (−0.993 + 0.111i)27-s + (0.467 − 0.884i)28-s + (−1.35 − 0.781i)31-s + (0.900 − 0.433i)36-s + (0.119 + 0.273i)37-s + (−0.563 + 1.82i)39-s + ⋯
L(s)  = 1  + (−0.999 + 0.0373i)3-s + (0.930 − 0.365i)4-s + (0.757 − 0.652i)7-s + (0.997 − 0.0747i)9-s + (−0.916 + 0.399i)12-s + (0.631 − 1.80i)13-s + (0.733 − 0.680i)16-s + (0.997 + 1.72i)19-s + (−0.733 + 0.680i)21-s + (−0.993 + 0.111i)27-s + (0.467 − 0.884i)28-s + (−1.35 − 0.781i)31-s + (0.900 − 0.433i)36-s + (0.119 + 0.273i)37-s + (−0.563 + 1.82i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $0.599 + 0.800i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ 0.599 + 0.800i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.367561352\)
\(L(\frac12)\) \(\approx\) \(1.367561352\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.999 - 0.0373i)T \)
5 \( 1 \)
7 \( 1 + (-0.757 + 0.652i)T \)
good2 \( 1 + (-0.930 + 0.365i)T^{2} \)
11 \( 1 + (0.988 + 0.149i)T^{2} \)
13 \( 1 + (-0.631 + 1.80i)T + (-0.781 - 0.623i)T^{2} \)
17 \( 1 + (-0.294 - 0.955i)T^{2} \)
19 \( 1 + (-0.997 - 1.72i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.294 - 0.955i)T^{2} \)
29 \( 1 + (-0.222 + 0.974i)T^{2} \)
31 \( 1 + (1.35 + 0.781i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.119 - 0.273i)T + (-0.680 + 0.733i)T^{2} \)
41 \( 1 + (-0.900 - 0.433i)T^{2} \)
43 \( 1 + (0.461 - 0.734i)T + (-0.433 - 0.900i)T^{2} \)
47 \( 1 + (0.930 - 0.365i)T^{2} \)
53 \( 1 + (0.680 + 0.733i)T^{2} \)
59 \( 1 + (-0.826 - 0.563i)T^{2} \)
61 \( 1 + (1.73 + 0.680i)T + (0.733 + 0.680i)T^{2} \)
67 \( 1 + (-0.352 - 1.31i)T + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + (0.222 + 0.974i)T^{2} \)
73 \( 1 + (0.367 - 1.94i)T + (-0.930 - 0.365i)T^{2} \)
79 \( 1 + (-0.632 + 0.365i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.781 + 0.623i)T^{2} \)
89 \( 1 + (0.988 - 0.149i)T^{2} \)
97 \( 1 + (1.16 + 1.16i)T + iT^{2} \)
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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

−8.264395365758638313578623133795, −7.68367055598114986399560879977, −7.21984519333490395029454711803, −6.15100418945684037878248003085, −5.66861740376115394979373133026, −5.14438500936660741139006893028, −3.96901156136649027292002620219, −3.17123623553280963820870881730, −1.70700916271263280156799866559, −1.00533971897390543905609758253, 1.45590475992361019992616660324, 2.12549319249093900302283080598, 3.34502678891710479915470615265, 4.40442045608437421491468427114, 5.10868590821708314214529491547, 5.90344782206013568694718415405, 6.69576583994693552457816335940, 7.12332667889630496719452451094, 7.87141138306554170984458836517, 9.019939423321913556589015192350

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