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Properties

Label 2-3675-735.134-c0-0-0
Degree $2$
Conductor $3675$
Sign $0.970 + 0.239i$
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  + (−0.781 + 0.623i)3-s + (0.900 − 0.433i)4-s + (−0.974 + 0.222i)7-s + (0.222 − 0.974i)9-s + (−0.433 + 0.900i)12-s + (1.21 − 0.277i)13-s + (0.623 − 0.781i)16-s + 0.445·19-s + (0.623 − 0.781i)21-s + (0.433 + 0.900i)27-s + (−0.781 + 0.623i)28-s − 1.80·31-s + (−0.222 − 0.974i)36-s + (0.781 − 1.62i)37-s + (−0.777 + 0.974i)39-s + ⋯
L(s)  = 1  + (−0.781 + 0.623i)3-s + (0.900 − 0.433i)4-s + (−0.974 + 0.222i)7-s + (0.222 − 0.974i)9-s + (−0.433 + 0.900i)12-s + (1.21 − 0.277i)13-s + (0.623 − 0.781i)16-s + 0.445·19-s + (0.623 − 0.781i)21-s + (0.433 + 0.900i)27-s + (−0.781 + 0.623i)28-s − 1.80·31-s + (−0.222 − 0.974i)36-s + (0.781 − 1.62i)37-s + (−0.777 + 0.974i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $0.970 + 0.239i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (3074, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ 0.970 + 0.239i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.145361343\)
\(L(\frac12)\) \(\approx\) \(1.145361343\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.781 - 0.623i)T \)
5 \( 1 \)
7 \( 1 + (0.974 - 0.222i)T \)
good2 \( 1 + (-0.900 + 0.433i)T^{2} \)
11 \( 1 + (0.900 - 0.433i)T^{2} \)
13 \( 1 + (-1.21 + 0.277i)T + (0.900 - 0.433i)T^{2} \)
17 \( 1 + (0.623 + 0.781i)T^{2} \)
19 \( 1 - 0.445T + T^{2} \)
23 \( 1 + (0.623 - 0.781i)T^{2} \)
29 \( 1 + (-0.623 - 0.781i)T^{2} \)
31 \( 1 + 1.80T + T^{2} \)
37 \( 1 + (-0.781 + 1.62i)T + (-0.623 - 0.781i)T^{2} \)
41 \( 1 + (0.222 - 0.974i)T^{2} \)
43 \( 1 + (0.347 + 0.277i)T + (0.222 + 0.974i)T^{2} \)
47 \( 1 + (-0.900 + 0.433i)T^{2} \)
53 \( 1 + (0.623 - 0.781i)T^{2} \)
59 \( 1 + (0.222 + 0.974i)T^{2} \)
61 \( 1 + (-1.62 - 0.781i)T + (0.623 + 0.781i)T^{2} \)
67 \( 1 + 1.24iT - T^{2} \)
71 \( 1 + (-0.623 + 0.781i)T^{2} \)
73 \( 1 + (-1.75 - 0.400i)T + (0.900 + 0.433i)T^{2} \)
79 \( 1 - 1.80T + T^{2} \)
83 \( 1 + (-0.900 - 0.433i)T^{2} \)
89 \( 1 + (0.900 + 0.433i)T^{2} \)
97 \( 1 - 0.445iT - T^{2} \)
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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

−8.992858655234923147118490914795, −7.79323679627107221069597191887, −6.92777778769452134532315626514, −6.36688204533143210485151866272, −5.69294043055104713965201295108, −5.26488162237710418662722358986, −3.84124827142551312044129125898, −3.40378760890795448920727447770, −2.19550364119851383400742822471, −0.846499857595137543132850576068, 1.14911348129700714778075886134, 2.17657286651140783568337501724, 3.25028080476704169459403519648, 3.93620682540076022051901588265, 5.21296131120159672822114069283, 6.04515292893072934055612383555, 6.53756241667373059972926223923, 7.07267574419607877483016770457, 7.82792878344980159720324237581, 8.529166264265977400340458189191

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