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Properties

Label 2-63e2-441.94-c0-0-0
Degree $2$
Conductor $3969$
Sign $0.943 + 0.331i$
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  + (0.988 + 0.149i)4-s + (−0.826 − 0.563i)7-s + (0.587 + 0.0440i)13-s + (0.955 + 0.294i)16-s + (0.751 − 0.433i)19-s + (−0.900 − 0.433i)25-s + (−0.733 − 0.680i)28-s + (1.61 − 0.930i)31-s + (−0.266 + 0.680i)37-s + (0.109 − 0.101i)43-s + (0.365 + 0.930i)49-s + (0.574 + 0.131i)52-s + (−0.0444 − 0.294i)61-s + (0.900 + 0.433i)64-s + (0.733 + 1.26i)67-s + ⋯
L(s)  = 1  + (0.988 + 0.149i)4-s + (−0.826 − 0.563i)7-s + (0.587 + 0.0440i)13-s + (0.955 + 0.294i)16-s + (0.751 − 0.433i)19-s + (−0.900 − 0.433i)25-s + (−0.733 − 0.680i)28-s + (1.61 − 0.930i)31-s + (−0.266 + 0.680i)37-s + (0.109 − 0.101i)43-s + (0.365 + 0.930i)49-s + (0.574 + 0.131i)52-s + (−0.0444 − 0.294i)61-s + (0.900 + 0.433i)64-s + (0.733 + 1.26i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $0.943 + 0.331i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (3916, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ 0.943 + 0.331i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.597155799\)
\(L(\frac12)\) \(\approx\) \(1.597155799\)
\(L(1)\) 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 \( 1 + (-0.988 - 0.149i)T^{2} \)
5 \( 1 + (0.900 + 0.433i)T^{2} \)
11 \( 1 + (0.623 - 0.781i)T^{2} \)
13 \( 1 + (-0.587 - 0.0440i)T + (0.988 + 0.149i)T^{2} \)
17 \( 1 + (-0.955 + 0.294i)T^{2} \)
19 \( 1 + (-0.751 + 0.433i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.222 - 0.974i)T^{2} \)
29 \( 1 + (-0.733 - 0.680i)T^{2} \)
31 \( 1 + (-1.61 + 0.930i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.266 - 0.680i)T + (-0.733 - 0.680i)T^{2} \)
41 \( 1 + (-0.0747 - 0.997i)T^{2} \)
43 \( 1 + (-0.109 + 0.101i)T + (0.0747 - 0.997i)T^{2} \)
47 \( 1 + (0.988 + 0.149i)T^{2} \)
53 \( 1 + (-0.733 + 0.680i)T^{2} \)
59 \( 1 + (-0.0747 + 0.997i)T^{2} \)
61 \( 1 + (0.0444 + 0.294i)T + (-0.955 + 0.294i)T^{2} \)
67 \( 1 + (-0.733 - 1.26i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.222 - 0.974i)T^{2} \)
73 \( 1 + (0.880 + 1.29i)T + (-0.365 + 0.930i)T^{2} \)
79 \( 1 + (-0.988 + 1.71i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.988 - 0.149i)T^{2} \)
89 \( 1 + (0.988 - 0.149i)T^{2} \)
97 \( 1 + (-1.72 + 0.997i)T + (0.5 - 0.866i)T^{2} \)
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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

−8.465578753837057377830053777172, −7.71562836460122580090732790805, −7.15275555478171526359642045299, −6.28177227547813329793603406528, −6.03047166040469668233084827518, −4.79188921623807426109029347319, −3.80411201592501623127989730185, −3.13253335457994735922877956456, −2.26902817144006696465041027767, −1.01744317676668136906087102444, 1.25744009273168013388596891797, 2.34967935571848838356399775726, 3.14444073466813658071294192202, 3.84723100419871575123878983561, 5.18841238064445295777472761448, 5.83911836344011592910640512238, 6.44566730394412774823978662334, 7.08694306587427110572199008181, 7.917402197783068770690311450325, 8.596142872357926642063530382969

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