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Properties

Label 2-63e2-441.292-c0-0-0
Degree $2$
Conductor $3969$
Sign $0.777 + 0.629i$
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  + (0.900 − 0.433i)4-s + (0.733 − 0.680i)7-s + (0.0878 − 0.284i)13-s + (0.623 − 0.781i)16-s + (1.68 + 0.974i)19-s + (−0.733 − 0.680i)25-s + (0.365 − 0.930i)28-s + 1.12i·31-s + (−1.36 + 0.930i)37-s + (−1.44 − 0.218i)43-s + (0.0747 − 0.997i)49-s + (−0.0444 − 0.294i)52-s + (0.865 − 1.79i)61-s + (0.222 − 0.974i)64-s + 0.730·67-s + ⋯
L(s)  = 1  + (0.900 − 0.433i)4-s + (0.733 − 0.680i)7-s + (0.0878 − 0.284i)13-s + (0.623 − 0.781i)16-s + (1.68 + 0.974i)19-s + (−0.733 − 0.680i)25-s + (0.365 − 0.930i)28-s + 1.12i·31-s + (−1.36 + 0.930i)37-s + (−1.44 − 0.218i)43-s + (0.0747 − 0.997i)49-s + (−0.0444 − 0.294i)52-s + (0.865 − 1.79i)61-s + (0.222 − 0.974i)64-s + 0.730·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $0.777 + 0.629i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (3673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ 0.777 + 0.629i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.835155834\)
\(L(\frac12)\) \(\approx\) \(1.835155834\)
\(L(1)\) 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 \( 1 + (-0.733 + 0.680i)T \)
good2 \( 1 + (-0.900 + 0.433i)T^{2} \)
5 \( 1 + (0.733 + 0.680i)T^{2} \)
11 \( 1 + (0.826 + 0.563i)T^{2} \)
13 \( 1 + (-0.0878 + 0.284i)T + (-0.826 - 0.563i)T^{2} \)
17 \( 1 + (0.988 - 0.149i)T^{2} \)
19 \( 1 + (-1.68 - 0.974i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.365 + 0.930i)T^{2} \)
29 \( 1 + (-0.988 + 0.149i)T^{2} \)
31 \( 1 - 1.12iT - T^{2} \)
37 \( 1 + (1.36 - 0.930i)T + (0.365 - 0.930i)T^{2} \)
41 \( 1 + (-0.955 + 0.294i)T^{2} \)
43 \( 1 + (1.44 + 0.218i)T + (0.955 + 0.294i)T^{2} \)
47 \( 1 + (0.900 - 0.433i)T^{2} \)
53 \( 1 + (0.365 + 0.930i)T^{2} \)
59 \( 1 + (0.222 + 0.974i)T^{2} \)
61 \( 1 + (-0.865 + 1.79i)T + (-0.623 - 0.781i)T^{2} \)
67 \( 1 - 0.730T + T^{2} \)
71 \( 1 + (0.623 - 0.781i)T^{2} \)
73 \( 1 + (-0.255 - 0.829i)T + (-0.826 + 0.563i)T^{2} \)
79 \( 1 - 0.149T + T^{2} \)
83 \( 1 + (-0.826 + 0.563i)T^{2} \)
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

−8.180731386812672329131607906339, −7.941400091585489918075001495707, −6.98517780473728797822711711402, −6.54645808325138898849540750055, −5.37366744679395720552276122787, −5.13289597142563274990765295030, −3.80799433613815957331934575977, −3.12246078315828412576367924593, −1.89027554782593363911365266694, −1.17210555900211459284678047767, 1.46846682725775958676270420625, 2.31712399606218457416427949611, 3.15547385909937658057464627882, 4.03044255142276102796785490960, 5.19071406006177010484410791332, 5.64151523502466743236576526542, 6.63699214235545553882929957572, 7.33622632209051949348134335980, 7.84659548264461296289707999811, 8.661340250038709039605639714015

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