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Properties

Label 2-3675-735.167-c0-0-1
Degree $2$
Conductor $3675$
Sign $-0.566 + 0.824i$
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  + (0.532 − 0.846i)3-s + (−0.781 + 0.623i)4-s + (−0.943 + 0.330i)7-s + (−0.433 − 0.900i)9-s + (0.111 + 0.993i)12-s + (−0.146 − 0.420i)13-s + (0.222 − 0.974i)16-s + 0.867·19-s + (−0.222 + 0.974i)21-s + (−0.993 − 0.111i)27-s + (0.532 − 0.846i)28-s − 1.56i·31-s + (0.900 + 0.433i)36-s + (−1.55 + 0.175i)37-s + (−0.433 − 0.0990i)39-s + ⋯
L(s)  = 1  + (0.532 − 0.846i)3-s + (−0.781 + 0.623i)4-s + (−0.943 + 0.330i)7-s + (−0.433 − 0.900i)9-s + (0.111 + 0.993i)12-s + (−0.146 − 0.420i)13-s + (0.222 − 0.974i)16-s + 0.867·19-s + (−0.222 + 0.974i)21-s + (−0.993 − 0.111i)27-s + (0.532 − 0.846i)28-s − 1.56i·31-s + (0.900 + 0.433i)36-s + (−1.55 + 0.175i)37-s + (−0.433 − 0.0990i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-0.566 + 0.824i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (3107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ -0.566 + 0.824i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7067030959\)
\(L(\frac12)\) \(\approx\) \(0.7067030959\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.532 + 0.846i)T \)
5 \( 1 \)
7 \( 1 + (0.943 - 0.330i)T \)
good2 \( 1 + (0.781 - 0.623i)T^{2} \)
11 \( 1 + (-0.623 - 0.781i)T^{2} \)
13 \( 1 + (0.146 + 0.420i)T + (-0.781 + 0.623i)T^{2} \)
17 \( 1 + (0.974 - 0.222i)T^{2} \)
19 \( 1 - 0.867T + T^{2} \)
23 \( 1 + (-0.974 - 0.222i)T^{2} \)
29 \( 1 + (-0.222 - 0.974i)T^{2} \)
31 \( 1 + 1.56iT - T^{2} \)
37 \( 1 + (1.55 - 0.175i)T + (0.974 - 0.222i)T^{2} \)
41 \( 1 + (-0.900 + 0.433i)T^{2} \)
43 \( 1 + (0.461 + 0.734i)T + (-0.433 + 0.900i)T^{2} \)
47 \( 1 + (-0.781 + 0.623i)T^{2} \)
53 \( 1 + (-0.974 - 0.222i)T^{2} \)
59 \( 1 + (0.900 + 0.433i)T^{2} \)
61 \( 1 + (1.22 + 0.974i)T + (0.222 + 0.974i)T^{2} \)
67 \( 1 + (1.37 + 1.37i)T + iT^{2} \)
71 \( 1 + (0.222 - 0.974i)T^{2} \)
73 \( 1 + (1.17 + 0.411i)T + (0.781 + 0.623i)T^{2} \)
79 \( 1 + 1.24iT - T^{2} \)
83 \( 1 + (-0.781 - 0.623i)T^{2} \)
89 \( 1 + (-0.623 + 0.781i)T^{2} \)
97 \( 1 + (-1.27 + 1.27i)T - iT^{2} \)
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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

−8.512443406028146650106326948946, −7.62904329568574339856689946169, −7.26954492583483279728093473351, −6.25874831448387742602832609502, −5.59200056296400200684890219534, −4.55989293658825024485364325809, −3.36537915832786174539287742105, −3.17987874116460189989861208854, −1.97576096905418113196058318914, −0.39089081646855797356337877290, 1.45235807419803501004625709180, 2.88336470718647220378239637265, 3.58602167205224706761199158483, 4.36886450424124535642032236849, 5.09868959606504079398628153317, 5.78562867457717566329939724617, 6.74839597229506266537618098329, 7.53841179762748853056847647206, 8.589894080073800327515205112582, 9.034492519860816883052872316078

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