Properties

Label 2-63e2-9.2-c0-0-5
Degree $2$
Conductor $3969$
Sign $0.642 - 0.766i$
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  + (1.22 + 0.707i)2-s + (0.499 + 0.866i)4-s + (1.22 + 0.707i)11-s + (0.499 − 0.866i)16-s + (0.999 + 1.73i)22-s + (1.22 − 0.707i)23-s + (−0.5 + 0.866i)25-s + (−1.22 − 0.707i)29-s + (1.22 − 0.707i)32-s + 1.41i·44-s + 2·46-s + (−1.22 + 0.707i)50-s + 1.41i·53-s + (−0.999 − 1.73i)58-s + 0.999·64-s + ⋯
L(s)  = 1  + (1.22 + 0.707i)2-s + (0.499 + 0.866i)4-s + (1.22 + 0.707i)11-s + (0.499 − 0.866i)16-s + (0.999 + 1.73i)22-s + (1.22 − 0.707i)23-s + (−0.5 + 0.866i)25-s + (−1.22 − 0.707i)29-s + (1.22 − 0.707i)32-s + 1.41i·44-s + 2·46-s + (−1.22 + 0.707i)50-s + 1.41i·53-s + (−0.999 − 1.73i)58-s + 0.999·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (1079, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ 0.642 - 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.695915578\)
\(L(\frac12)\) \(\approx\) \(2.695915578\)
\(L(1)\) 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 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 - 1.41iT - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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

−8.708356089389508588028396085083, −7.57725904956757378037439286546, −7.11298044439842804612560330565, −6.45139157630485489602570697506, −5.74277810231563740501728061195, −5.01174706373990138068642246131, −4.20968621606679618683479706587, −3.72032912777597919451117200989, −2.65480485602388637514979764527, −1.35139518479425626988002304017, 1.29770229535788770028691411624, 2.26912644634595336716858864484, 3.40225787720413641132118707418, 3.69995649107587175395581497799, 4.67162470091720355594567084568, 5.38080501808663299775779592343, 6.11043012879546794792683810892, 6.80556909808827812053212292081, 7.76442278236509571835162000772, 8.675793022444090429711915764680

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