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Properties

Label 2-3900-195.179-c0-0-3
Degree $2$
Conductor $3900$
Sign $0.0471 + 0.998i$
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  + (0.866 − 0.5i)3-s + (0.866 − 1.5i)7-s + (0.499 − 0.866i)9-s + (−0.866 − 0.5i)13-s − 1.73i·21-s − 0.999i·27-s + 1.73i·31-s − 0.999·39-s + (−0.866 − 0.5i)43-s + (−1 − 1.73i)49-s + (−0.5 + 0.866i)61-s + (−0.866 − 1.49i)63-s + (0.866 + 1.5i)67-s + 1.73·73-s − 79-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)3-s + (0.866 − 1.5i)7-s + (0.499 − 0.866i)9-s + (−0.866 − 0.5i)13-s − 1.73i·21-s − 0.999i·27-s + 1.73i·31-s − 0.999·39-s + (−0.866 − 0.5i)43-s + (−1 − 1.73i)49-s + (−0.5 + 0.866i)61-s + (−0.866 − 1.49i)63-s + (0.866 + 1.5i)67-s + 1.73·73-s − 79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0471 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0471 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3900\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.0471 + 0.998i$
Analytic conductor: \(1.94635\)
Root analytic conductor: \(1.39511\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3900} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3900,\ (\ :0),\ 0.0471 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.849162188\)
\(L(\frac12)\) \(\approx\) \(1.849162188\)
\(L(1)\) 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 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
13 \( 1 + (0.866 + 0.5i)T \)
good7 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T^{2} \)
73 \( 1 - 1.73T + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.320865159737940404050795114866, −7.72466977241103583266219574302, −7.14514236429775462362233359261, −6.65632023485159544983022265168, −5.33923408183111877642187229569, −4.58163645309557950153266039395, −3.79358007908885200334710726614, −2.98037074350049797110761642678, −1.88403008670882000735334378703, −0.963932451966779567588631025558, 1.89056619586825361440576518877, 2.34557860682055104755866488647, 3.28640540643186846391984677673, 4.38936364391784012356348090518, 4.95947590840783585223800437179, 5.68036405010341390087595240078, 6.65611637905187079376165219138, 7.73252076265150356876667580764, 8.101110697338723578631523162131, 8.865848185555504615118983419811

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