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Properties

Label 2-63e2-9.5-c0-0-7
Degree $2$
Conductor $3969$
Sign $-0.573 + 0.819i$
Analytic cond. $1.98078$
Root an. cond. $1.40740$
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  + (1.67 − 0.965i)2-s + (1.36 − 2.36i)4-s − 3.34i·8-s + (−0.448 + 0.258i)11-s + (−1.86 − 3.23i)16-s + (−0.500 + 0.866i)22-s + (−1.22 − 0.707i)23-s + (−0.5 − 0.866i)25-s + (1.22 − 0.707i)29-s + (−3.34 − 1.93i)32-s + 1.73·37-s + (0.866 + 1.5i)43-s + 1.41i·44-s − 2.73·46-s + (−1.67 − 0.965i)50-s + ⋯
L(s)  = 1  + (1.67 − 0.965i)2-s + (1.36 − 2.36i)4-s − 3.34i·8-s + (−0.448 + 0.258i)11-s + (−1.86 − 3.23i)16-s + (−0.500 + 0.866i)22-s + (−1.22 − 0.707i)23-s + (−0.5 − 0.866i)25-s + (1.22 − 0.707i)29-s + (−3.34 − 1.93i)32-s + 1.73·37-s + (0.866 + 1.5i)43-s + 1.41i·44-s − 2.73·46-s + (−1.67 − 0.965i)50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $-0.573 + 0.819i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (2402, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ -0.573 + 0.819i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.304178404\)
\(L(\frac12)\) \(\approx\) \(3.304178404\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-1.67 + 0.965i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.448 - 0.258i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - 1.73T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - 0.517iT - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - 1.93iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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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

−8.282017402145279487586112994403, −7.49425733427074326875384445538, −6.30140128697986479517984408523, −6.15621927786749810998489940904, −5.15961582410821253722747454319, −4.34219161788439275490870241999, −4.01960535654412021917288555314, −2.66336652511246627045139889142, −2.44576478758211864214911083990, −1.09157854206959105930273808817, 2.00792654727085090795069357868, 2.98494539481738082362262050106, 3.69959504800961879500979198901, 4.46807498521896846663461597003, 5.22507379227390733570251307754, 5.87802783798510307817693422263, 6.41945326673668596714198493438, 7.35265957482370487828774316381, 7.78721500730702890571173075475, 8.487688442801203863187780177043

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