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Properties

Label 2-63e2-441.29-c0-0-0
Degree $2$
Conductor $3969$
Sign $-0.184 + 0.982i$
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  + (0.365 − 0.930i)4-s + (0.826 − 0.563i)7-s + (−0.0332 − 0.443i)13-s + (−0.733 − 0.680i)16-s − 1.80·19-s + (0.0747 − 0.997i)25-s + (−0.222 − 0.974i)28-s + (−0.623 − 1.07i)31-s + (0.777 − 0.974i)37-s + (1.32 + 1.22i)43-s + (0.365 − 0.930i)49-s + (−0.425 − 0.131i)52-s + (0.455 + 1.16i)61-s + (−0.900 + 0.433i)64-s + (0.222 + 0.385i)67-s + ⋯
L(s)  = 1  + (0.365 − 0.930i)4-s + (0.826 − 0.563i)7-s + (−0.0332 − 0.443i)13-s + (−0.733 − 0.680i)16-s − 1.80·19-s + (0.0747 − 0.997i)25-s + (−0.222 − 0.974i)28-s + (−0.623 − 1.07i)31-s + (0.777 − 0.974i)37-s + (1.32 + 1.22i)43-s + (0.365 − 0.930i)49-s + (−0.425 − 0.131i)52-s + (0.455 + 1.16i)61-s + (−0.900 + 0.433i)64-s + (0.222 + 0.385i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.184 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.184 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $-0.184 + 0.982i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (1646, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ -0.184 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.373468462\)
\(L(\frac12)\) \(\approx\) \(1.373468462\)
\(L(1)\) 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 \( 1 + (-0.826 + 0.563i)T \)
good2 \( 1 + (-0.365 + 0.930i)T^{2} \)
5 \( 1 + (-0.0747 + 0.997i)T^{2} \)
11 \( 1 + (-0.365 + 0.930i)T^{2} \)
13 \( 1 + (0.0332 + 0.443i)T + (-0.988 + 0.149i)T^{2} \)
17 \( 1 + (0.222 + 0.974i)T^{2} \)
19 \( 1 + 1.80T + T^{2} \)
23 \( 1 + (0.733 + 0.680i)T^{2} \)
29 \( 1 + (0.733 - 0.680i)T^{2} \)
31 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.777 + 0.974i)T + (-0.222 - 0.974i)T^{2} \)
41 \( 1 + (-0.0747 + 0.997i)T^{2} \)
43 \( 1 + (-1.32 - 1.22i)T + (0.0747 + 0.997i)T^{2} \)
47 \( 1 + (-0.365 + 0.930i)T^{2} \)
53 \( 1 + (0.222 - 0.974i)T^{2} \)
59 \( 1 + (-0.826 + 0.563i)T^{2} \)
61 \( 1 + (-0.455 - 1.16i)T + (-0.733 + 0.680i)T^{2} \)
67 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.222 - 0.974i)T^{2} \)
73 \( 1 + (1.12 - 0.541i)T + (0.623 - 0.781i)T^{2} \)
79 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.988 + 0.149i)T^{2} \)
89 \( 1 + (-0.623 + 0.781i)T^{2} \)
97 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.401764790685815036755483585158, −7.65539923969771552892489518734, −6.97368124113945369626559453281, −6.07864459285490922675909283530, −5.63482185517600106354370191333, −4.49015326550186015588155239476, −4.18201662316961447464638992415, −2.62500702372428722872590671518, −1.92211688384069197610249959417, −0.74336020448828270466700451196, 1.73026660442835358051843578700, 2.40179473453467344091406319593, 3.42456508196723480935484946189, 4.28115530494469773265587263476, 4.97195558754033618536243011104, 5.99038707752070490441024239723, 6.71364657779054879522502533247, 7.46106555938181995381433175575, 8.116782304600358331566991432281, 8.818092405988726007396490377171

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