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Properties

Label 2-14e2-196.103-c1-0-14
Degree $2$
Conductor $196$
Sign $0.450 - 0.892i$
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  + (0.976 + 1.02i)2-s + (0.268 + 0.182i)3-s + (−0.0944 + 1.99i)4-s + (3.60 + 0.269i)5-s + (0.0746 + 0.453i)6-s + (−1.91 − 1.82i)7-s + (−2.13 + 1.85i)8-s + (−1.05 − 2.69i)9-s + (3.23 + 3.94i)10-s + (0.946 + 0.371i)11-s + (−0.390 + 0.518i)12-s + (−2.72 + 2.17i)13-s + (−0.00528 − 3.74i)14-s + (0.916 + 0.731i)15-s + (−3.98 − 0.377i)16-s + (−1.83 − 1.97i)17-s + ⋯
L(s)  = 1  + (0.690 + 0.723i)2-s + (0.154 + 0.105i)3-s + (−0.0472 + 0.998i)4-s + (1.61 + 0.120i)5-s + (0.0304 + 0.184i)6-s + (−0.724 − 0.689i)7-s + (−0.755 + 0.655i)8-s + (−0.352 − 0.898i)9-s + (1.02 + 1.24i)10-s + (0.285 + 0.111i)11-s + (−0.112 + 0.149i)12-s + (−0.755 + 0.602i)13-s + (−0.00141 − 0.999i)14-s + (0.236 + 0.188i)15-s + (−0.995 − 0.0943i)16-s + (−0.444 − 0.478i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.450 - 0.892i$
Analytic conductor: \(1.56506\)
Root analytic conductor: \(1.25102\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :1/2),\ 0.450 - 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60791 + 0.989640i\)
\(L(\frac12)\) \(\approx\) \(1.60791 + 0.989640i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.976 - 1.02i)T \)
7 \( 1 + (1.91 + 1.82i)T \)
good3 \( 1 + (-0.268 - 0.182i)T + (1.09 + 2.79i)T^{2} \)
5 \( 1 + (-3.60 - 0.269i)T + (4.94 + 0.745i)T^{2} \)
11 \( 1 + (-0.946 - 0.371i)T + (8.06 + 7.48i)T^{2} \)
13 \( 1 + (2.72 - 2.17i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + (1.83 + 1.97i)T + (-1.27 + 16.9i)T^{2} \)
19 \( 1 + (2.18 - 3.77i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.02 + 4.33i)T + (-1.71 - 22.9i)T^{2} \)
29 \( 1 + (-1.26 - 5.55i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 + (2.93 + 5.08i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.33 + 0.721i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (2.01 + 4.17i)T + (-25.5 + 32.0i)T^{2} \)
43 \( 1 + (-1.13 + 2.35i)T + (-26.8 - 33.6i)T^{2} \)
47 \( 1 + (-1.76 + 0.266i)T + (44.9 - 13.8i)T^{2} \)
53 \( 1 + (-7.54 + 2.32i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (-1.04 - 13.8i)T + (-58.3 + 8.79i)T^{2} \)
61 \( 1 + (1.60 - 5.20i)T + (-50.4 - 34.3i)T^{2} \)
67 \( 1 + (-12.0 + 6.98i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.12 + 1.39i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (-0.391 + 2.59i)T + (-69.7 - 21.5i)T^{2} \)
79 \( 1 + (-8.77 - 5.06i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.43 - 11.8i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (3.57 - 1.40i)T + (65.2 - 60.5i)T^{2} \)
97 \( 1 - 13.8iT - 97T^{2} \)
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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

−12.90605551865408042486213457427, −12.11311132598404250569090002429, −10.57155845436452298637546452544, −9.485837337230696867553608952294, −8.867486137738589577053088167096, −7.01739435734067323248781915406, −6.49953154186960013313300943659, −5.44097897469576626846341071406, −4.00640824814492293688146882031, −2.57351785610940353612962619274, 2.02866093142643035613051914594, 2.92822293000588688095801486217, 4.99142988789807192240941241977, 5.72802583020417240798556614232, 6.73465515125069556027795424107, 8.748366573622457618811712970633, 9.578489364491978753162043109759, 10.35130198947635444540001396009, 11.36742591424558317096897097778, 12.71818857668034559558842990233

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