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Properties

Label 2-14e2-28.19-c1-0-5
Degree $2$
Conductor $196$
Sign $0.378 + 0.925i$
Analytic cond. $1.56506$
Root an. cond. $1.25102$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.895 − 1.09i)2-s + (−0.395 + 1.96i)4-s + (2.49 − 1.32i)8-s + (1.5 − 2.59i)9-s + (4.58 − 2.64i)11-s + (−3.68 − 1.55i)16-s + (−4.18 + 0.685i)18-s + (−6.99 − 2.64i)22-s + (4.58 + 2.64i)23-s + (−2.5 − 4.33i)25-s − 2·29-s + (1.60 + 5.42i)32-s + (4.49 + 3.96i)36-s + (−3 + 5.19i)37-s + 5.29i·43-s + (3.37 + 10.0i)44-s + ⋯
L(s)  = 1  + (−0.633 − 0.773i)2-s + (−0.197 + 0.980i)4-s + (0.883 − 0.467i)8-s + (0.5 − 0.866i)9-s + (1.38 − 0.797i)11-s + (−0.921 − 0.387i)16-s + (−0.986 + 0.161i)18-s + (−1.49 − 0.564i)22-s + (0.955 + 0.551i)23-s + (−0.5 − 0.866i)25-s − 0.371·29-s + (0.283 + 0.958i)32-s + (0.749 + 0.661i)36-s + (−0.493 + 0.854i)37-s + 0.806i·43-s + (0.508 + 1.51i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.378 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.378 + 0.925i$
Analytic conductor: \(1.56506\)
Root analytic conductor: \(1.25102\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :1/2),\ 0.378 + 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.767031 - 0.515214i\)
\(L(\frac12)\) \(\approx\) \(0.767031 - 0.515214i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.895 + 1.09i)T \)
7 \( 1 \)
good3 \( 1 + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.58 + 2.64i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.58 - 2.64i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3 - 5.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 5.29iT - 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5 - 8.66i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (13.7 - 7.93i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.29iT - 71T^{2} \)
73 \( 1 + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (13.7 + 7.93i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−12.00264385996560793849805593791, −11.47526629447669567014353971579, −10.29868440891307741296997628749, −9.317930915324724375829408442346, −8.681365177359837863344972974138, −7.32128405161204637453610980480, −6.24717727928567446695323503089, −4.27950580911791396004361692953, −3.23756362214674834415442174929, −1.23845354689855811952550514282, 1.72936483391171059479635891350, 4.23739907177150787592050740914, 5.40592446642738935757152614136, 6.79721347299415402208623122425, 7.44159961406715001642149827690, 8.719858549194704313227725266593, 9.551978801375808694968419197358, 10.50327735275171279047341053341, 11.50877349618274784214652833200, 12.81471602609523602339860749154

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