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Properties

Label 2-3675-735.83-c0-0-1
Degree $2$
Conductor $3675$
Sign $0.455 + 0.890i$
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  + (−0.111 − 0.993i)3-s + (−0.433 + 0.900i)4-s + (−0.846 + 0.532i)7-s + (−0.974 + 0.222i)9-s + (0.943 + 0.330i)12-s + (−0.663 − 1.05i)13-s + (−0.623 − 0.781i)16-s + 1.94·19-s + (0.623 + 0.781i)21-s + (0.330 + 0.943i)27-s + (−0.111 − 0.993i)28-s + 0.867i·31-s + (0.222 − 0.974i)36-s + (0.286 − 0.819i)37-s + (−0.974 + 0.777i)39-s + ⋯
L(s)  = 1  + (−0.111 − 0.993i)3-s + (−0.433 + 0.900i)4-s + (−0.846 + 0.532i)7-s + (−0.974 + 0.222i)9-s + (0.943 + 0.330i)12-s + (−0.663 − 1.05i)13-s + (−0.623 − 0.781i)16-s + 1.94·19-s + (0.623 + 0.781i)21-s + (0.330 + 0.943i)27-s + (−0.111 − 0.993i)28-s + 0.867i·31-s + (0.222 − 0.974i)36-s + (0.286 − 0.819i)37-s + (−0.974 + 0.777i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.455 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.455 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $0.455 + 0.890i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (818, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ 0.455 + 0.890i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8081999570\)
\(L(\frac12)\) \(\approx\) \(0.8081999570\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.111 + 0.993i)T \)
5 \( 1 \)
7 \( 1 + (0.846 - 0.532i)T \)
good2 \( 1 + (0.433 - 0.900i)T^{2} \)
11 \( 1 + (0.900 + 0.433i)T^{2} \)
13 \( 1 + (0.663 + 1.05i)T + (-0.433 + 0.900i)T^{2} \)
17 \( 1 + (-0.781 - 0.623i)T^{2} \)
19 \( 1 - 1.94T + T^{2} \)
23 \( 1 + (0.781 - 0.623i)T^{2} \)
29 \( 1 + (0.623 - 0.781i)T^{2} \)
31 \( 1 - 0.867iT - T^{2} \)
37 \( 1 + (-0.286 + 0.819i)T + (-0.781 - 0.623i)T^{2} \)
41 \( 1 + (-0.222 - 0.974i)T^{2} \)
43 \( 1 + (-0.218 + 1.93i)T + (-0.974 - 0.222i)T^{2} \)
47 \( 1 + (-0.433 + 0.900i)T^{2} \)
53 \( 1 + (0.781 - 0.623i)T^{2} \)
59 \( 1 + (0.222 - 0.974i)T^{2} \)
61 \( 1 + (0.376 + 0.781i)T + (-0.623 + 0.781i)T^{2} \)
67 \( 1 + (-1.10 + 1.10i)T - iT^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (-1.52 - 0.958i)T + (0.433 + 0.900i)T^{2} \)
79 \( 1 + 1.80iT - T^{2} \)
83 \( 1 + (-0.433 - 0.900i)T^{2} \)
89 \( 1 + (0.900 - 0.433i)T^{2} \)
97 \( 1 + (-0.314 - 0.314i)T + iT^{2} \)
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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

−8.492546581823759921443947353612, −7.67196113635982098270484047445, −7.31846956997657997923887353615, −6.51008533631719994761751070974, −5.47895508482102040874427828002, −5.10971095733327993936332103454, −3.56362569799785029506440203506, −3.07834696533054645425530718625, −2.24271115752319633351946668810, −0.58536439411565196086206876043, 1.02992279943480553107589807325, 2.58423857976384076359424966819, 3.55640053542429815251537768933, 4.36443287155322552165820567731, 4.96378983459316100759118029521, 5.77011551739020832364823035166, 6.45053274936258422548450500010, 7.26297993451767991924229487748, 8.250292438583062013870963708600, 9.337128531616750328170155985157

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