Properties

Label 2-63e2-441.220-c0-0-0
Degree $2$
Conductor $3969$
Sign $-0.515 + 0.856i$
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  + (−0.0747 − 0.997i)4-s + (−0.955 + 0.294i)7-s + (0.202 − 0.218i)13-s + (−0.988 + 0.149i)16-s + (1.68 − 0.974i)19-s + (−0.222 − 0.974i)25-s + (0.365 + 0.930i)28-s + (−0.975 + 0.563i)31-s + (−1.36 − 0.930i)37-s + (0.535 − 1.36i)43-s + (0.826 − 0.563i)49-s + (−0.233 − 0.185i)52-s + (−1.98 − 0.149i)61-s + (0.222 + 0.974i)64-s + (−0.365 − 0.632i)67-s + ⋯
L(s)  = 1  + (−0.0747 − 0.997i)4-s + (−0.955 + 0.294i)7-s + (0.202 − 0.218i)13-s + (−0.988 + 0.149i)16-s + (1.68 − 0.974i)19-s + (−0.222 − 0.974i)25-s + (0.365 + 0.930i)28-s + (−0.975 + 0.563i)31-s + (−1.36 − 0.930i)37-s + (0.535 − 1.36i)43-s + (0.826 − 0.563i)49-s + (−0.233 − 0.185i)52-s + (−1.98 − 0.149i)61-s + (0.222 + 0.974i)64-s + (−0.365 − 0.632i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $-0.515 + 0.856i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (514, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ -0.515 + 0.856i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8586878070\)
\(L(\frac12)\) \(\approx\) \(0.8586878070\)
\(L(1)\) 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 \( 1 + (0.955 - 0.294i)T \)
good2 \( 1 + (0.0747 + 0.997i)T^{2} \)
5 \( 1 + (0.222 + 0.974i)T^{2} \)
11 \( 1 + (-0.900 - 0.433i)T^{2} \)
13 \( 1 + (-0.202 + 0.218i)T + (-0.0747 - 0.997i)T^{2} \)
17 \( 1 + (0.988 + 0.149i)T^{2} \)
19 \( 1 + (-1.68 + 0.974i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.623 + 0.781i)T^{2} \)
29 \( 1 + (0.365 + 0.930i)T^{2} \)
31 \( 1 + (0.975 - 0.563i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (1.36 + 0.930i)T + (0.365 + 0.930i)T^{2} \)
41 \( 1 + (0.733 - 0.680i)T^{2} \)
43 \( 1 + (-0.535 + 1.36i)T + (-0.733 - 0.680i)T^{2} \)
47 \( 1 + (-0.0747 - 0.997i)T^{2} \)
53 \( 1 + (0.365 - 0.930i)T^{2} \)
59 \( 1 + (0.733 + 0.680i)T^{2} \)
61 \( 1 + (1.98 + 0.149i)T + (0.988 + 0.149i)T^{2} \)
67 \( 1 + (0.365 + 0.632i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.623 + 0.781i)T^{2} \)
73 \( 1 + (-0.255 + 0.829i)T + (-0.826 - 0.563i)T^{2} \)
79 \( 1 + (0.0747 - 0.129i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.0747 + 0.997i)T^{2} \)
89 \( 1 + (-0.0747 + 0.997i)T^{2} \)
97 \( 1 + (-1.17 + 0.680i)T + (0.5 - 0.866i)T^{2} \)
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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

−8.662903546883766706285106917482, −7.42487013737595577094152894544, −6.91953187249337585411733709237, −6.05326887505621461120306893310, −5.50171851986667669116617821780, −4.80895434787952981075929125780, −3.68551371530232781036469534303, −2.86704678295177389637665995430, −1.80804860237661146562377715402, −0.48548408706461466005693011345, 1.47600724528512333738501594933, 2.87684814488145382697501710325, 3.44812596508248330712675424465, 4.06802637434885839041657117167, 5.15535094388898651561127360625, 5.98746592638345907415110483098, 6.83225348530948740477983314727, 7.50687404737827032531848484927, 7.947570689903981180235446186482, 9.022374164217805751941205946052

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