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Properties

Label 2-63e2-63.34-c0-0-0
Degree $2$
Conductor $3969$
Sign $-0.513 - 0.858i$
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  + (0.5 + 0.866i)4-s + (−1.5 + 0.866i)13-s + (−0.499 + 0.866i)16-s + (−0.5 + 0.866i)25-s + (−1.5 + 0.866i)31-s + 37-s + (−0.5 + 0.866i)43-s + (−1.5 − 0.866i)52-s + (1.5 + 0.866i)61-s − 0.999·64-s + (0.5 + 0.866i)67-s + (0.5 − 0.866i)79-s + (−1.5 − 0.866i)97-s − 0.999·100-s + (1.5 − 0.866i)103-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)4-s + (−1.5 + 0.866i)13-s + (−0.499 + 0.866i)16-s + (−0.5 + 0.866i)25-s + (−1.5 + 0.866i)31-s + 37-s + (−0.5 + 0.866i)43-s + (−1.5 − 0.866i)52-s + (1.5 + 0.866i)61-s − 0.999·64-s + (0.5 + 0.866i)67-s + (0.5 − 0.866i)79-s + (−1.5 − 0.866i)97-s − 0.999·100-s + (1.5 − 0.866i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.513 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.513 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $-0.513 - 0.858i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (2890, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ -0.513 - 0.858i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.044064701\)
\(L(\frac12)\) \(\approx\) \(1.044064701\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
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 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
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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

−8.858561275579786176919822847537, −8.045445164091596667332800159124, −7.23213558758402373801545868619, −7.03776479185730429533245104601, −6.00592548546053580085476194099, −5.06983926954906560621920157010, −4.26529569216343145408406655274, −3.43911606352665801079611925090, −2.54082095977327984818975988398, −1.74831525948880254976969904946, 0.52969452594193309368424775607, 2.01296345732541481641191206378, 2.59421546092086221886333336536, 3.76458866592458228603940365871, 4.84434974216422494233187513194, 5.41140664579752667003013632863, 6.11285843152788242766111171673, 6.95320869621390275724149568373, 7.57149871812589374236894713967, 8.259930802173647595907153822480

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