Properties

Label 2-63e2-441.349-c0-0-0
Degree $2$
Conductor $3969$
Sign $-0.368 + 0.929i$
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  + (0.733 − 0.680i)4-s + (−0.365 − 0.930i)7-s + (0.129 − 0.858i)13-s + (0.0747 − 0.997i)16-s − 1.56i·19-s + (−0.988 + 0.149i)25-s + (−0.900 − 0.433i)28-s + (−1.68 + 0.974i)31-s + (−0.0990 + 0.433i)37-s + (−0.0931 + 1.24i)43-s + (−0.733 + 0.680i)49-s + (−0.488 − 0.716i)52-s + (1.32 − 1.42i)61-s + (−0.623 − 0.781i)64-s + (0.900 + 1.56i)67-s + ⋯
L(s)  = 1  + (0.733 − 0.680i)4-s + (−0.365 − 0.930i)7-s + (0.129 − 0.858i)13-s + (0.0747 − 0.997i)16-s − 1.56i·19-s + (−0.988 + 0.149i)25-s + (−0.900 − 0.433i)28-s + (−1.68 + 0.974i)31-s + (−0.0990 + 0.433i)37-s + (−0.0931 + 1.24i)43-s + (−0.733 + 0.680i)49-s + (−0.488 − 0.716i)52-s + (1.32 − 1.42i)61-s + (−0.623 − 0.781i)64-s + (0.900 + 1.56i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $-0.368 + 0.929i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ -0.368 + 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.290880845\)
\(L(\frac12)\) \(\approx\) \(1.290880845\)
\(L(1)\) 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 \( 1 + (-0.733 + 0.680i)T^{2} \)
5 \( 1 + (0.988 - 0.149i)T^{2} \)
11 \( 1 + (-0.733 + 0.680i)T^{2} \)
13 \( 1 + (-0.129 + 0.858i)T + (-0.955 - 0.294i)T^{2} \)
17 \( 1 + (0.900 + 0.433i)T^{2} \)
19 \( 1 + 1.56iT - T^{2} \)
23 \( 1 + (0.0747 - 0.997i)T^{2} \)
29 \( 1 + (0.0747 + 0.997i)T^{2} \)
31 \( 1 + (1.68 - 0.974i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.0990 - 0.433i)T + (-0.900 - 0.433i)T^{2} \)
41 \( 1 + (0.988 - 0.149i)T^{2} \)
43 \( 1 + (0.0931 - 1.24i)T + (-0.988 - 0.149i)T^{2} \)
47 \( 1 + (0.733 - 0.680i)T^{2} \)
53 \( 1 + (-0.900 + 0.433i)T^{2} \)
59 \( 1 + (-0.365 - 0.930i)T^{2} \)
61 \( 1 + (-1.32 + 1.42i)T + (-0.0747 - 0.997i)T^{2} \)
67 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.900 + 0.433i)T^{2} \)
73 \( 1 + (-1.52 + 1.21i)T + (0.222 - 0.974i)T^{2} \)
79 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.955 + 0.294i)T^{2} \)
89 \( 1 + (0.222 - 0.974i)T^{2} \)
97 \( 1 + (1.35 + 0.781i)T + (0.5 + 0.866i)T^{2} \)
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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

−8.309322813289233466668309193434, −7.46556189111908310808541419821, −6.96077962347703303525868647364, −6.30593412334511679103367581403, −5.44071031624139126383186635668, −4.78561831180863013055812971789, −3.64520164619256640969827947963, −2.91265413117410159890176988567, −1.82104246668330316746685665588, −0.67194179367339255774335175128, 1.88472467868362839842850309857, 2.31939791160533708681465709112, 3.64012054096661683027890120605, 3.90139445567513073242783589177, 5.37699675910743127373718830884, 5.95324445722341032196108830122, 6.65203552108404917852207039597, 7.43314886984000585536203097645, 8.098911459223988923292134372644, 8.783676084892009497317962582486

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