Properties

Label 2-14e2-28.19-c1-0-15
Degree $2$
Conductor $196$
Sign $-0.690 - 0.723i$
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  + (−0.0345 − 1.41i)2-s + (−1.36 − 2.35i)3-s + (−1.99 + 0.0976i)4-s + (−0.937 − 0.541i)5-s + (−3.28 + 2.00i)6-s + (0.207 + 2.82i)8-s + (−2.20 + 3.82i)9-s + (−0.732 + 1.34i)10-s + (1.80 − 1.04i)11-s + (2.94 + 4.57i)12-s − 2.61i·13-s + 2.94i·15-s + (3.98 − 0.390i)16-s + (−3.86 + 2.23i)17-s + (5.48 + 2.98i)18-s + (0.563 − 0.976i)19-s + ⋯
L(s)  = 1  + (−0.0244 − 0.999i)2-s + (−0.786 − 1.36i)3-s + (−0.998 + 0.0488i)4-s + (−0.419 − 0.242i)5-s + (−1.34 + 0.819i)6-s + (0.0732 + 0.997i)8-s + (−0.735 + 1.27i)9-s + (−0.231 + 0.424i)10-s + (0.544 − 0.314i)11-s + (0.851 + 1.32i)12-s − 0.724i·13-s + 0.760i·15-s + (0.995 − 0.0975i)16-s + (−0.936 + 0.540i)17-s + (1.29 + 0.704i)18-s + (0.129 − 0.224i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.690 - 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $-0.690 - 0.723i$
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: \((2,\ 196,\ (\ :1/2),\ -0.690 - 0.723i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.216242 + 0.505331i\)
\(L(\frac12)\) \(\approx\) \(0.216242 + 0.505331i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0345 + 1.41i)T \)
7 \( 1 \)
good3 \( 1 + (1.36 + 2.35i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.937 + 0.541i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.80 + 1.04i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.61iT - 13T^{2} \)
17 \( 1 + (3.86 - 2.23i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.563 + 0.976i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.16 + 3.55i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.17T + 29T^{2} \)
31 \( 1 + (-3.85 - 6.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.54iT - 41T^{2} \)
43 \( 1 + 7.97iT - 43T^{2} \)
47 \( 1 + (-2.72 + 4.71i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.24 - 5.61i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.41 + 7.64i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-11.3 - 6.53i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.71 + 5.03i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-6.90 + 3.98i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.60 - 2.08i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.31T + 83T^{2} \)
89 \( 1 + (3.47 + 2.00i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.82iT - 97T^{2} \)
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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

−12.13371921895823136184604687705, −11.16634941202724975138383202371, −10.30933463459180731545616809162, −8.741804564656705696557365032177, −7.958816301135228164758572417905, −6.62158673988568725578763936652, −5.49963888500103844072091198371, −4.01617488843726946961716657791, −2.14934716165382407392882569475, −0.54402811388778689484603540861, 3.88951453544464495392766063749, 4.54932568026663069429187552370, 5.76686871677544479735721597354, 6.75487576908697859345966421251, 8.034109940122487941383878573307, 9.464557905846740464485514471779, 9.767191082659790996540161357211, 11.21117293982924204449187393663, 11.81680348945903227915963638178, 13.33677789157423951729566111718

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