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Properties

Label 2-63e2-441.304-c0-0-0
Degree $2$
Conductor $3969$
Sign $-0.620 - 0.784i$
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  + (0.222 + 0.974i)4-s + (−0.365 − 0.930i)7-s + (−0.297 + 1.97i)13-s + (−0.900 + 0.433i)16-s + (−1.35 + 0.781i)19-s + (0.365 − 0.930i)25-s + (0.826 − 0.563i)28-s + 0.589i·31-s + (−1.82 + 0.563i)37-s + (−0.0546 + 0.728i)43-s + (−0.733 + 0.680i)49-s + (−1.98 + 0.149i)52-s + (1.32 + 0.302i)61-s + (−0.623 − 0.781i)64-s + 1.65·67-s + ⋯
L(s)  = 1  + (0.222 + 0.974i)4-s + (−0.365 − 0.930i)7-s + (−0.297 + 1.97i)13-s + (−0.900 + 0.433i)16-s + (−1.35 + 0.781i)19-s + (0.365 − 0.930i)25-s + (0.826 − 0.563i)28-s + 0.589i·31-s + (−1.82 + 0.563i)37-s + (−0.0546 + 0.728i)43-s + (−0.733 + 0.680i)49-s + (−1.98 + 0.149i)52-s + (1.32 + 0.302i)61-s + (−0.623 − 0.781i)64-s + 1.65·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.620 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.620 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $-0.620 - 0.784i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (3538, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ -0.620 - 0.784i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8491339512\)
\(L(\frac12)\) \(\approx\) \(0.8491339512\)
\(L(1)\) 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 \( 1 + (0.365 + 0.930i)T \)
good2 \( 1 + (-0.222 - 0.974i)T^{2} \)
5 \( 1 + (-0.365 + 0.930i)T^{2} \)
11 \( 1 + (0.955 + 0.294i)T^{2} \)
13 \( 1 + (0.297 - 1.97i)T + (-0.955 - 0.294i)T^{2} \)
17 \( 1 + (-0.0747 - 0.997i)T^{2} \)
19 \( 1 + (1.35 - 0.781i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.826 + 0.563i)T^{2} \)
29 \( 1 + (0.0747 + 0.997i)T^{2} \)
31 \( 1 - 0.589iT - T^{2} \)
37 \( 1 + (1.82 - 0.563i)T + (0.826 - 0.563i)T^{2} \)
41 \( 1 + (0.988 - 0.149i)T^{2} \)
43 \( 1 + (0.0546 - 0.728i)T + (-0.988 - 0.149i)T^{2} \)
47 \( 1 + (0.222 + 0.974i)T^{2} \)
53 \( 1 + (0.826 + 0.563i)T^{2} \)
59 \( 1 + (-0.623 + 0.781i)T^{2} \)
61 \( 1 + (-1.32 - 0.302i)T + (0.900 + 0.433i)T^{2} \)
67 \( 1 - 1.65T + T^{2} \)
71 \( 1 + (-0.900 + 0.433i)T^{2} \)
73 \( 1 + (-0.290 - 1.92i)T + (-0.955 + 0.294i)T^{2} \)
79 \( 1 + 1.46T + T^{2} \)
83 \( 1 + (-0.955 + 0.294i)T^{2} \)
89 \( 1 + (0.733 + 0.680i)T^{2} \)
97 \( 1 + (-1.61 - 0.930i)T + (0.5 + 0.866i)T^{2} \)
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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

−8.628803212577953689267501501418, −8.327427226003140634323336795003, −7.24580487534744868439195035015, −6.77475415199006943463268467181, −6.35270847419347799482863232861, −4.89633241778651316173784730531, −4.12250454981637963034245387967, −3.73563150176303343266960992736, −2.56739494023270630183016171024, −1.67081107853750593970545667347, 0.43963459911706736213690792734, 1.95297044228519951975089663064, 2.68941510415299302864919279222, 3.61611358563654612579927853971, 5.00149255329485039473688346735, 5.34606834896637775170143360208, 6.07582154753323648634790507106, 6.77967525891873255778617503069, 7.58598013487366822938313981210, 8.589926215392064471476238545685

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