Properties

Label 2-63e2-63.2-c0-0-3
Degree $2$
Conductor $3969$
Sign $0.954 + 0.296i$
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 − 1.41i·11-s + (0.499 + 0.866i)16-s + (1.00 + 1.73i)22-s + 1.41i·23-s + 25-s + (−1.22 − 0.707i)29-s + (−1.22 − 0.707i)32-s + (−1.22 − 0.707i)44-s + (−1.00 − 1.73i)46-s + (−1.22 + 0.707i)50-s + (1.22 − 0.707i)53-s + 2·58-s + 0.999·64-s + ⋯
L(s)  = 1  + (−1.22 + 0.707i)2-s + (0.499 − 0.866i)4-s − 1.41i·11-s + (0.499 + 0.866i)16-s + (1.00 + 1.73i)22-s + 1.41i·23-s + 25-s + (−1.22 − 0.707i)29-s + (−1.22 − 0.707i)32-s + (−1.22 − 0.707i)44-s + (−1.00 − 1.73i)46-s + (−1.22 + 0.707i)50-s + (1.22 − 0.707i)53-s + 2·58-s + 0.999·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $0.954 + 0.296i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (3509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ 0.954 + 0.296i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5821774161\)
\(L(\frac12)\) \(\approx\) \(0.5821774161\)
\(L(1)\) 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 \( 1 - T^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 - 1.41iT - T^{2} \)
29 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
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 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
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

−8.672957281809207667144861596026, −7.84282387628413785786680644402, −7.41815542299176498850375685173, −6.49947191463816452622312322505, −5.91509545421500459905379293666, −5.16374134524701658411467613424, −3.85379958417569450328375084324, −3.18974392243905934361556201746, −1.76577461545252241622351190108, −0.58779138915295839115903035492, 1.09195187999228471846735585429, 2.12283783571082670787491667006, 2.75948145275648521759044180893, 3.99701773176668348038927274079, 4.85539147311880201060511145956, 5.66186782385849580269429033617, 6.94704489609035353981663165135, 7.21952114621616392550246030328, 8.236134793495151317261377067565, 8.762745365432376273881345224011

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