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Properties

Label 2-63e2-63.40-c0-0-0
Degree $2$
Conductor $3969$
Sign $-0.959 - 0.281i$
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  − 1.73·2-s + 1.99·4-s − 1.73·8-s + (−0.866 + 1.5i)11-s + 0.999·16-s + (1.49 − 2.59i)22-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)37-s + (−0.5 − 0.866i)43-s + (−1.73 + 2.99i)44-s + (0.866 − 1.49i)50-s + (−0.866 − 1.5i)53-s − 1.00·64-s − 67-s − 1.73·71-s + ⋯
L(s)  = 1  − 1.73·2-s + 1.99·4-s − 1.73·8-s + (−0.866 + 1.5i)11-s + 0.999·16-s + (1.49 − 2.59i)22-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)37-s + (−0.5 − 0.866i)43-s + (−1.73 + 2.99i)44-s + (0.866 − 1.49i)50-s + (−0.866 − 1.5i)53-s − 1.00·64-s − 67-s − 1.73·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $-0.959 - 0.281i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (3106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ -0.959 - 0.281i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1650830480\)
\(L(\frac12)\) \(\approx\) \(0.1650830480\)
\(L(1)\) 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 \( 1 + 1.73T + T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 + 1.73T + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + T + T^{2} \)
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

−8.966794242989773844597192046195, −8.296600660636332606671011342826, −7.55095778280558648896240248324, −7.18808986247612027813289758780, −6.40259389998765960586485167931, −5.34950153356990217512012478059, −4.52058801246556881069772529681, −3.21796899667575957517255585707, −2.19591301989554076321134192568, −1.52747262569354730044111210592, 0.16490124962486815178949979230, 1.38415930238760546561510544364, 2.52256023423446477942133308056, 3.23081534851153694707841920922, 4.51473506307316178015408403540, 5.74103451688420627806200466782, 6.19315876370603071433018479095, 7.18675593022836852909732183192, 7.81686435203158666107771485012, 8.387118692861016460009771669974

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