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Properties

Label 2-3675-735.167-c0-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(0.9913075300\)
\(L(\frac12)\) \(\approx\) \(0.9913075300\)
\(L(1)\) 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

−8.983022142285744544695822094138, −8.012376326954251370994478030372, −7.64059270107424993167484685355, −6.51537620741523119766427223897, −5.59108100238637372122100549772, −4.91345417506433114379718648738, −4.24016680932365614857612906899, −3.72754689545111479374031948029, −2.59114966475035870552273565608, −0.942869656853906178624000728263, 0.931637026062635663073736213317, 1.73576785858545561680201428446, 2.91134637803879635881647063332, 4.23575776481815271308960707728, 5.09044764225575424863541236538, 5.48428653205646442553530232630, 6.23534797329550086545917331738, 7.14019552618185917167409427428, 7.958434963137560041868260775879, 8.440877486374553092256867661392

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