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Properties

Label 2-63e2-441.221-c0-0-0
Degree $2$
Conductor $3969$
Sign $0.515 - 0.856i$
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  + (0.0747 + 0.997i)4-s + (0.955 − 0.294i)7-s + (1.44 + 1.34i)13-s + (−0.988 + 0.149i)16-s + (0.222 + 0.385i)19-s + (−0.222 − 0.974i)25-s + (0.365 + 0.930i)28-s + (−0.826 − 1.43i)31-s + (1.36 + 0.930i)37-s + (−0.535 + 1.36i)43-s + (0.826 − 0.563i)49-s + (−1.23 + 1.54i)52-s + (0.0111 − 0.149i)61-s + (−0.222 − 0.974i)64-s + (−0.365 − 0.632i)67-s + ⋯
L(s)  = 1  + (0.0747 + 0.997i)4-s + (0.955 − 0.294i)7-s + (1.44 + 1.34i)13-s + (−0.988 + 0.149i)16-s + (0.222 + 0.385i)19-s + (−0.222 − 0.974i)25-s + (0.365 + 0.930i)28-s + (−0.826 − 1.43i)31-s + (1.36 + 0.930i)37-s + (−0.535 + 1.36i)43-s + (0.826 − 0.563i)49-s + (−1.23 + 1.54i)52-s + (0.0111 − 0.149i)61-s + (−0.222 − 0.974i)64-s + (−0.365 − 0.632i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $0.515 - 0.856i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (3455, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ 0.515 - 0.856i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.560474875\)
\(L(\frac12)\) \(\approx\) \(1.560474875\)
\(L(1)\) 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 \( 1 + (-0.955 + 0.294i)T \)
good2 \( 1 + (-0.0747 - 0.997i)T^{2} \)
5 \( 1 + (0.222 + 0.974i)T^{2} \)
11 \( 1 + (0.900 + 0.433i)T^{2} \)
13 \( 1 + (-1.44 - 1.34i)T + (0.0747 + 0.997i)T^{2} \)
17 \( 1 + (0.988 + 0.149i)T^{2} \)
19 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.623 - 0.781i)T^{2} \)
29 \( 1 + (-0.365 - 0.930i)T^{2} \)
31 \( 1 + (0.826 + 1.43i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-1.36 - 0.930i)T + (0.365 + 0.930i)T^{2} \)
41 \( 1 + (0.733 - 0.680i)T^{2} \)
43 \( 1 + (0.535 - 1.36i)T + (-0.733 - 0.680i)T^{2} \)
47 \( 1 + (-0.0747 - 0.997i)T^{2} \)
53 \( 1 + (-0.365 + 0.930i)T^{2} \)
59 \( 1 + (0.733 + 0.680i)T^{2} \)
61 \( 1 + (-0.0111 + 0.149i)T + (-0.988 - 0.149i)T^{2} \)
67 \( 1 + (0.365 + 0.632i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (1.72 + 0.531i)T + (0.826 + 0.563i)T^{2} \)
79 \( 1 + (0.0747 - 0.129i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.0747 + 0.997i)T^{2} \)
89 \( 1 + (-0.0747 + 0.997i)T^{2} \)
97 \( 1 + (-0.733 - 1.26i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.651423771145054492985253298723, −7.954306015748536089068397264713, −7.54429305009033486248092271192, −6.51117693305505548994197469567, −6.01923791686650984899281343233, −4.67603983515175918660608267305, −4.20761918063658746331385719071, −3.48879122142635661260406646198, −2.33193907483568875118936152149, −1.41901709168753492119246052855, 1.01264632104681792651876881871, 1.81273673711692265371062995121, 2.97507928312188684646742339862, 3.96447533918926105484802364773, 5.01219810966357614837305589702, 5.55567839184929762627892477979, 6.04065035493012404953481821052, 7.10230582251545920543636119266, 7.75879622912600900309683477460, 8.765250259962288445597185054001

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