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Properties

Label 2-63e2-441.200-c0-0-0
Degree $2$
Conductor $3969$
Sign $0.772 + 0.634i$
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  + (0.623 − 0.781i)4-s + (0.0747 − 0.997i)7-s + (1.57 + 1.07i)13-s + (−0.222 − 0.974i)16-s + (0.900 + 1.56i)19-s + (0.0747 + 0.997i)25-s + (−0.733 − 0.680i)28-s + 0.730·31-s + (0.266 − 0.680i)37-s + (0.142 + 0.0440i)43-s + (−0.988 − 0.149i)49-s + (1.82 − 0.563i)52-s + (−1.23 − 1.54i)61-s + (−0.900 − 0.433i)64-s − 1.46·67-s + ⋯
L(s)  = 1  + (0.623 − 0.781i)4-s + (0.0747 − 0.997i)7-s + (1.57 + 1.07i)13-s + (−0.222 − 0.974i)16-s + (0.900 + 1.56i)19-s + (0.0747 + 0.997i)25-s + (−0.733 − 0.680i)28-s + 0.730·31-s + (0.266 − 0.680i)37-s + (0.142 + 0.0440i)43-s + (−0.988 − 0.149i)49-s + (1.82 − 0.563i)52-s + (−1.23 − 1.54i)61-s + (−0.900 − 0.433i)64-s − 1.46·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $0.772 + 0.634i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (2699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ 0.772 + 0.634i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.700217936\)
\(L(\frac12)\) \(\approx\) \(1.700217936\)
\(L(1)\) 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 \( 1 + (-0.0747 + 0.997i)T \)
good2 \( 1 + (-0.623 + 0.781i)T^{2} \)
5 \( 1 + (-0.0747 - 0.997i)T^{2} \)
11 \( 1 + (-0.365 - 0.930i)T^{2} \)
13 \( 1 + (-1.57 - 1.07i)T + (0.365 + 0.930i)T^{2} \)
17 \( 1 + (-0.955 + 0.294i)T^{2} \)
19 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.733 - 0.680i)T^{2} \)
29 \( 1 + (-0.955 + 0.294i)T^{2} \)
31 \( 1 - 0.730T + T^{2} \)
37 \( 1 + (-0.266 + 0.680i)T + (-0.733 - 0.680i)T^{2} \)
41 \( 1 + (-0.826 + 0.563i)T^{2} \)
43 \( 1 + (-0.142 - 0.0440i)T + (0.826 + 0.563i)T^{2} \)
47 \( 1 + (-0.623 + 0.781i)T^{2} \)
53 \( 1 + (0.733 - 0.680i)T^{2} \)
59 \( 1 + (0.900 - 0.433i)T^{2} \)
61 \( 1 + (1.23 + 1.54i)T + (-0.222 + 0.974i)T^{2} \)
67 \( 1 + 1.46T + T^{2} \)
71 \( 1 + (0.222 + 0.974i)T^{2} \)
73 \( 1 + (-1.03 + 0.702i)T + (0.365 - 0.930i)T^{2} \)
79 \( 1 + 1.97T + T^{2} \)
83 \( 1 + (-0.365 + 0.930i)T^{2} \)
89 \( 1 + (0.988 - 0.149i)T^{2} \)
97 \( 1 + (0.0747 - 0.129i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.533868027243137104971958566914, −7.65379861130969805352048514312, −7.09668887525946369317244087654, −6.23791252074279107233465087769, −5.84063454421532562399083683763, −4.78657587519526753994331762581, −3.90735244746185743265613477353, −3.20454186314667887812640029309, −1.71772003279302345609541340712, −1.22685535397770199833614127525, 1.27938204874974313350473804496, 2.71946151405560196917343636805, 2.93419417810210247412322178405, 4.03545032428749597457012604082, 5.01818341609627086272080247799, 5.93582316846199143205959504396, 6.41387485368171594821392228402, 7.31981718498850007699220293400, 8.095545565084361626197374350824, 8.597321037428701035126385165708

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