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Properties

Label 2-3900-39.17-c0-0-1
Degree $2$
Conductor $3900$
Sign $0.872 + 0.488i$
Analytic cond. $1.94635$
Root an. cond. $1.39511$
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  + (−0.5 + 0.866i)3-s + (1.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (−0.5 − 0.866i)13-s + 1.73i·21-s + 0.999·27-s − 1.73i·31-s + 0.999·39-s + (−0.5 − 0.866i)43-s + (1 − 1.73i)49-s + (−0.5 − 0.866i)61-s + (−1.49 − 0.866i)63-s + (−1.5 − 0.866i)67-s + 1.73i·73-s + 79-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (1.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (−0.5 − 0.866i)13-s + 1.73i·21-s + 0.999·27-s − 1.73i·31-s + 0.999·39-s + (−0.5 − 0.866i)43-s + (1 − 1.73i)49-s + (−0.5 − 0.866i)61-s + (−1.49 − 0.866i)63-s + (−1.5 − 0.866i)67-s + 1.73i·73-s + 79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3900\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.872 + 0.488i$
Analytic conductor: \(1.94635\)
Root analytic conductor: \(1.39511\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3900} (2201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3900,\ (\ :0),\ 0.872 + 0.488i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.163882988\)
\(L(\frac12)\) \(\approx\) \(1.163882988\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 \)
13 \( 1 + (0.5 + 0.866i)T \)
good7 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + 1.73iT - T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T^{2} \)
73 \( 1 - 1.73iT - T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
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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

−8.542110338701584067208060037230, −7.84889661441451401450641101941, −7.29187515378254523844048717688, −6.21493439968440076627790540922, −5.41034102742218248203758602912, −4.79863040313449921597909654137, −4.19262448857279291946003653549, −3.34529082748182244011008970023, −2.08936237658604651612867074256, −0.74356348143088851825620426428, 1.41548591437889166465225779715, 2.01269409725709878325998354999, 2.98231908193455674608022565921, 4.53325056863303068014134721982, 4.97006486390279834276943741173, 5.73692029414857773841666661665, 6.51672680780500020233836647572, 7.30583673828586677914674074459, 7.903684806416389606595512708292, 8.639558908851636562435126705866

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