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Properties

Label 2-3675-735.83-c0-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(1.310026336\)
\(L(\frac12)\) \(\approx\) \(1.310026336\)
\(L(1)\) 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

−8.981229035242977853204157507873, −8.273024354237362149192146740052, −7.67583580315377525071717688728, −6.90687856041272411683001839207, −5.78467460306229547914993189525, −4.75622697232592827345573986646, −4.52348473121013938430938950840, −3.51684688174591067931877714283, −2.99849220262137318701250478352, −1.47198144397277742492265205206, 0.848599217895718404195427182134, 1.67819823758922722625567172270, 2.69555000179772553685648001686, 3.74961268705593050527721889861, 5.01304992133286903005860516343, 5.62986728769194344343204800938, 5.94736828833218227040358585137, 7.14137244744887024510659574629, 7.70639326785727269122046728100, 8.513997738188452187569942807339

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