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Properties

Label 2-3675-735.593-c0-0-1
Degree $2$
Conductor $3675$
Sign $0.256 + 0.966i$
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  + (0.982 − 0.185i)3-s + (−0.294 − 0.955i)4-s + (0.916 − 0.399i)7-s + (0.930 − 0.365i)9-s + (−0.467 − 0.884i)12-s + (−0.148 − 0.0167i)13-s + (−0.826 + 0.563i)16-s + (0.930 − 1.61i)19-s + (0.826 − 0.563i)21-s + (0.846 − 0.532i)27-s + (−0.652 − 0.757i)28-s + (−1.68 + 0.974i)31-s + (−0.623 − 0.781i)36-s + (−1.20 + 0.635i)37-s + (−0.149 + 0.0111i)39-s + ⋯
L(s)  = 1  + (0.982 − 0.185i)3-s + (−0.294 − 0.955i)4-s + (0.916 − 0.399i)7-s + (0.930 − 0.365i)9-s + (−0.467 − 0.884i)12-s + (−0.148 − 0.0167i)13-s + (−0.826 + 0.563i)16-s + (0.930 − 1.61i)19-s + (0.826 − 0.563i)21-s + (0.846 − 0.532i)27-s + (−0.652 − 0.757i)28-s + (−1.68 + 0.974i)31-s + (−0.623 − 0.781i)36-s + (−1.20 + 0.635i)37-s + (−0.149 + 0.0111i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $0.256 + 0.966i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ 0.256 + 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.896329022\)
\(L(\frac12)\) \(\approx\) \(1.896329022\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.982 + 0.185i)T \)
5 \( 1 \)
7 \( 1 + (-0.916 + 0.399i)T \)
good2 \( 1 + (0.294 + 0.955i)T^{2} \)
11 \( 1 + (0.733 + 0.680i)T^{2} \)
13 \( 1 + (0.148 + 0.0167i)T + (0.974 + 0.222i)T^{2} \)
17 \( 1 + (-0.997 - 0.0747i)T^{2} \)
19 \( 1 + (-0.930 + 1.61i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.997 - 0.0747i)T^{2} \)
29 \( 1 + (-0.900 + 0.433i)T^{2} \)
31 \( 1 + (1.68 - 0.974i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (1.20 - 0.635i)T + (0.563 - 0.826i)T^{2} \)
41 \( 1 + (0.623 - 0.781i)T^{2} \)
43 \( 1 + (-0.516 - 1.47i)T + (-0.781 + 0.623i)T^{2} \)
47 \( 1 + (-0.294 - 0.955i)T^{2} \)
53 \( 1 + (-0.563 - 0.826i)T^{2} \)
59 \( 1 + (0.988 - 0.149i)T^{2} \)
61 \( 1 + (0.173 - 0.563i)T + (-0.826 - 0.563i)T^{2} \)
67 \( 1 + (-1.08 - 0.291i)T + (0.866 + 0.5i)T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (-1.17 + 0.870i)T + (0.294 - 0.955i)T^{2} \)
79 \( 1 + (1.65 + 0.955i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.974 - 0.222i)T^{2} \)
89 \( 1 + (0.733 - 0.680i)T^{2} \)
97 \( 1 + (-1.39 - 1.39i)T + iT^{2} \)
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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

−8.763345484218940744420945526021, −7.79438409531090853341842253860, −7.17769726017244833698006226247, −6.52605339022183764109828725443, −5.26624284722016390991442118202, −4.86472457430230759250110259895, −3.95932787859379961000231062403, −2.93566398952319537849779083383, −1.88405786112711866307091715149, −1.07763740568879346426323726955, 1.71588951909367196118192256005, 2.46604337689843105719013756711, 3.63104457538871599602135869908, 3.90429088348792156320772339281, 5.01197903120763785093116702777, 5.65857243110717499367600320597, 7.09429113452396450783052060045, 7.58471557071688174466207034648, 8.143088379511664112319039904287, 8.761668897184243748932833681819

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