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Properties

Label 2-14e2-28.23-c0-0-0
Degree $2$
Conductor $196$
Sign $0.386 + 0.922i$
Analytic cond. $0.0978167$
Root an. cond. $0.312756$
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s − 0.999·8-s + (−0.5 + 0.866i)9-s + (−0.5 + 0.866i)16-s + (0.499 + 0.866i)18-s + (0.5 + 0.866i)25-s − 2·29-s + (0.499 + 0.866i)32-s + 0.999·36-s + (1 − 1.73i)37-s + 0.999·50-s + (−1 − 1.73i)53-s + (−1 + 1.73i)58-s + 0.999·64-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s − 0.999·8-s + (−0.5 + 0.866i)9-s + (−0.5 + 0.866i)16-s + (0.499 + 0.866i)18-s + (0.5 + 0.866i)25-s − 2·29-s + (0.499 + 0.866i)32-s + 0.999·36-s + (1 − 1.73i)37-s + 0.999·50-s + (−1 − 1.73i)53-s + (−1 + 1.73i)58-s + 0.999·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(0.0978167\)
Root analytic conductor: \(0.312756\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :0),\ 0.386 + 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7969303608\)
\(L(\frac12)\) \(\approx\) \(0.7969303608\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
good3 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + 2T + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−12.69141897461786698404848434635, −11.35294802171642945672225285257, −10.97138784553999417208556347366, −9.756373681544834803200131627476, −8.820670258734472405030051147178, −7.49701636140976844582415404029, −5.89992524169561971065949613182, −4.97528652268026370505559725304, −3.58162411329461393066864114016, −2.12569062224151564491364255442, 3.09336987652443877185837786336, 4.38000068575935630123086381756, 5.72155615144303813595824092520, 6.58732678641571362341105103184, 7.74801773578209418145434274888, 8.782608087660897668507769907627, 9.670190919253859328223890927600, 11.23302451880094472070769875365, 12.16419199014233014976224479993, 13.00780039069378470022071841936

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