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Properties

Label 2-14e2-196.103-c1-0-12
Degree $2$
Conductor $196$
Sign $-0.455 + 0.890i$
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  + (−1.02 − 0.979i)2-s + (−1.63 − 1.11i)3-s + (0.0826 + 1.99i)4-s + (2.43 + 0.182i)5-s + (0.577 + 2.74i)6-s + (1.91 − 1.82i)7-s + (1.87 − 2.12i)8-s + (0.337 + 0.859i)9-s + (−2.30 − 2.57i)10-s + (−2.09 − 0.823i)11-s + (2.09 − 3.36i)12-s + (0.425 − 0.339i)13-s + (−3.74 − 0.00463i)14-s + (−3.78 − 3.01i)15-s + (−3.98 + 0.330i)16-s + (−2.31 − 2.49i)17-s + ⋯
L(s)  = 1  + (−0.721 − 0.692i)2-s + (−0.944 − 0.644i)3-s + (0.0413 + 0.999i)4-s + (1.08 + 0.0816i)5-s + (0.235 + 1.11i)6-s + (0.722 − 0.691i)7-s + (0.661 − 0.749i)8-s + (0.112 + 0.286i)9-s + (−0.729 − 0.813i)10-s + (−0.632 − 0.248i)11-s + (0.604 − 0.970i)12-s + (0.117 − 0.0940i)13-s + (−0.999 − 0.00123i)14-s + (−0.976 − 0.779i)15-s + (−0.996 + 0.0825i)16-s + (−0.562 − 0.606i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.455 + 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.455 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $-0.455 + 0.890i$
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: \((2,\ 196,\ (\ :1/2),\ -0.455 + 0.890i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.377315 - 0.616698i\)
\(L(\frac12)\) \(\approx\) \(0.377315 - 0.616698i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.02 + 0.979i)T \)
7 \( 1 + (-1.91 + 1.82i)T \)
good3 \( 1 + (1.63 + 1.11i)T + (1.09 + 2.79i)T^{2} \)
5 \( 1 + (-2.43 - 0.182i)T + (4.94 + 0.745i)T^{2} \)
11 \( 1 + (2.09 + 0.823i)T + (8.06 + 7.48i)T^{2} \)
13 \( 1 + (-0.425 + 0.339i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + (2.31 + 2.49i)T + (-1.27 + 16.9i)T^{2} \)
19 \( 1 + (0.309 - 0.535i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.40 + 5.82i)T + (-1.71 - 22.9i)T^{2} \)
29 \( 1 + (1.87 + 8.19i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 + (-0.779 - 1.35i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-11.4 - 3.53i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (0.189 + 0.393i)T + (-25.5 + 32.0i)T^{2} \)
43 \( 1 + (2.48 - 5.15i)T + (-26.8 - 33.6i)T^{2} \)
47 \( 1 + (2.28 - 0.344i)T + (44.9 - 13.8i)T^{2} \)
53 \( 1 + (-1.32 + 0.407i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (-0.851 - 11.3i)T + (-58.3 + 8.79i)T^{2} \)
61 \( 1 + (4.06 - 13.1i)T + (-50.4 - 34.3i)T^{2} \)
67 \( 1 + (5.00 - 2.89i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-9.15 - 2.08i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (-0.0617 + 0.409i)T + (-69.7 - 21.5i)T^{2} \)
79 \( 1 + (-5.72 - 3.30i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.68 + 7.12i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (-2.24 + 0.879i)T + (65.2 - 60.5i)T^{2} \)
97 \( 1 - 16.6iT - 97T^{2} \)
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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

−11.91035175257004534180725405793, −11.13257333211896226609910654397, −10.44504713951519309750733650479, −9.418188098331164778265459921714, −8.162180771656028572976941793899, −7.07404727994871619491724739312, −6.04477210281411513909401175764, −4.62331023946777626850552918624, −2.50963899366327486340994039708, −0.955856899306573965428602064407, 1.94539781430128658757605064105, 4.92097194907541784584189748568, 5.45284799855962929190333534901, 6.37110850401857206187662674471, 7.81681829774802727799471950698, 9.023257226873562151266832124272, 9.764013910537154760720441441503, 10.84151785131566691279195658692, 11.27853298618573098736480139493, 12.87349876005850017503366856722

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