Properties

Label 2-63e2-441.115-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  + (−0.623 − 0.781i)4-s + (−0.826 + 0.563i)7-s + (1.35 − 0.101i)13-s + (−0.222 + 0.974i)16-s + (−0.751 + 0.433i)19-s + (0.826 + 0.563i)25-s + (0.955 + 0.294i)28-s + 0.298i·31-s + (−1.95 − 0.294i)37-s + (1.21 + 1.12i)43-s + (0.365 − 0.930i)49-s + (−0.925 − 0.997i)52-s + (1.45 + 1.16i)61-s + (0.900 − 0.433i)64-s + 1.91·67-s + ⋯
L(s)  = 1  + (−0.623 − 0.781i)4-s + (−0.826 + 0.563i)7-s + (1.35 − 0.101i)13-s + (−0.222 + 0.974i)16-s + (−0.751 + 0.433i)19-s + (0.826 + 0.563i)25-s + (0.955 + 0.294i)28-s + 0.298i·31-s + (−1.95 − 0.294i)37-s + (1.21 + 1.12i)43-s + (0.365 − 0.930i)49-s + (−0.925 − 0.997i)52-s + (1.45 + 1.16i)61-s + (0.900 − 0.433i)64-s + 1.91·67-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: $\chi_{3969} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ 0.943 - 0.331i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9594872531\)
\(L(\frac12)\) \(\approx\) \(0.9594872531\)
\(L(1)\) 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 \( 1 + (0.623 + 0.781i)T^{2} \)
5 \( 1 + (-0.826 - 0.563i)T^{2} \)
11 \( 1 + (-0.988 + 0.149i)T^{2} \)
13 \( 1 + (-1.35 + 0.101i)T + (0.988 - 0.149i)T^{2} \)
17 \( 1 + (0.733 - 0.680i)T^{2} \)
19 \( 1 + (0.751 - 0.433i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.955 - 0.294i)T^{2} \)
29 \( 1 + (-0.733 + 0.680i)T^{2} \)
31 \( 1 - 0.298iT - T^{2} \)
37 \( 1 + (1.95 + 0.294i)T + (0.955 + 0.294i)T^{2} \)
41 \( 1 + (-0.0747 + 0.997i)T^{2} \)
43 \( 1 + (-1.21 - 1.12i)T + (0.0747 + 0.997i)T^{2} \)
47 \( 1 + (-0.623 - 0.781i)T^{2} \)
53 \( 1 + (0.955 - 0.294i)T^{2} \)
59 \( 1 + (0.900 + 0.433i)T^{2} \)
61 \( 1 + (-1.45 - 1.16i)T + (0.222 + 0.974i)T^{2} \)
67 \( 1 - 1.91T + T^{2} \)
71 \( 1 + (-0.222 + 0.974i)T^{2} \)
73 \( 1 + (-1.55 - 0.116i)T + (0.988 + 0.149i)T^{2} \)
79 \( 1 - 0.730T + T^{2} \)
83 \( 1 + (0.988 + 0.149i)T^{2} \)
89 \( 1 + (-0.365 - 0.930i)T^{2} \)
97 \( 1 + (0.975 + 0.563i)T + (0.5 + 0.866i)T^{2} \)
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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

−8.687698128050887106203145274130, −8.299067411686935366166132283646, −6.96674955636845298742671720300, −6.37304103092658818424055576402, −5.70654992238685699337719208896, −5.10579002648360073044272066428, −4.00788902912211600702274377418, −3.39937195653213960403518667072, −2.17053124571437284409861594508, −1.03490585043092686376852561667, 0.69088943876490711545441200411, 2.30424936780328246524353606565, 3.44861805060047863428875123172, 3.80234274850929067411759963178, 4.67472336356867048013048234778, 5.60826709574178446904257106681, 6.67132216248471957351084056593, 6.94192766185272304871905457900, 8.057627681155048186870791551727, 8.578926240271443796068944329228

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