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Properties

Label 2-63e2-441.239-c0-0-0
Degree $2$
Conductor $3969$
Sign $0.851 + 0.524i$
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  + (0.955 + 0.294i)4-s + (−0.988 − 0.149i)7-s + (−0.658 − 1.67i)13-s + (0.826 + 0.563i)16-s + 1.24·19-s + (0.365 − 0.930i)25-s + (−0.900 − 0.433i)28-s + (0.222 − 0.385i)31-s + (0.0990 − 0.433i)37-s + (1.03 + 0.702i)43-s + (0.955 + 0.294i)49-s + (−0.134 − 1.79i)52-s + (−0.425 + 0.131i)61-s + (0.623 + 0.781i)64-s + (0.900 − 1.56i)67-s + ⋯
L(s)  = 1  + (0.955 + 0.294i)4-s + (−0.988 − 0.149i)7-s + (−0.658 − 1.67i)13-s + (0.826 + 0.563i)16-s + 1.24·19-s + (0.365 − 0.930i)25-s + (−0.900 − 0.433i)28-s + (0.222 − 0.385i)31-s + (0.0990 − 0.433i)37-s + (1.03 + 0.702i)43-s + (0.955 + 0.294i)49-s + (−0.134 − 1.79i)52-s + (−0.425 + 0.131i)61-s + (0.623 + 0.781i)64-s + (0.900 − 1.56i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $0.851 + 0.524i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (1268, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ 0.851 + 0.524i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.455287834\)
\(L(\frac12)\) \(\approx\) \(1.455287834\)
\(L(1)\) 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 \( 1 + (0.988 + 0.149i)T \)
good2 \( 1 + (-0.955 - 0.294i)T^{2} \)
5 \( 1 + (-0.365 + 0.930i)T^{2} \)
11 \( 1 + (-0.955 - 0.294i)T^{2} \)
13 \( 1 + (0.658 + 1.67i)T + (-0.733 + 0.680i)T^{2} \)
17 \( 1 + (0.900 + 0.433i)T^{2} \)
19 \( 1 - 1.24T + T^{2} \)
23 \( 1 + (-0.826 - 0.563i)T^{2} \)
29 \( 1 + (-0.826 + 0.563i)T^{2} \)
31 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.0990 + 0.433i)T + (-0.900 - 0.433i)T^{2} \)
41 \( 1 + (-0.365 + 0.930i)T^{2} \)
43 \( 1 + (-1.03 - 0.702i)T + (0.365 + 0.930i)T^{2} \)
47 \( 1 + (-0.955 - 0.294i)T^{2} \)
53 \( 1 + (0.900 - 0.433i)T^{2} \)
59 \( 1 + (0.988 + 0.149i)T^{2} \)
61 \( 1 + (0.425 - 0.131i)T + (0.826 - 0.563i)T^{2} \)
67 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.900 - 0.433i)T^{2} \)
73 \( 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2} \)
79 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.733 + 0.680i)T^{2} \)
89 \( 1 + (0.222 - 0.974i)T^{2} \)
97 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.388467149206983180187350330038, −7.63212601854381098384103738705, −7.27801343983316549928418525477, −6.30556429736864692258154271541, −5.81856350244302655194761407476, −4.90917854971349468326665219709, −3.68312812915515800875959997910, −3.00328845263540959969064255759, −2.43102011012349232524326126668, −0.863164190853576811472320553346, 1.30885027836193699996835609299, 2.36374900731229685864484105466, 3.10175793396063602216324249696, 4.01859605792440987381374760047, 5.12330970725774536784798233628, 5.80584157529187782627980453004, 6.73129805563109034960185851339, 6.99465220896119875815332166785, 7.72389418853371288591877475082, 8.911001117834751917574931822008

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