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Properties

Label 2-14e2-196.111-c1-0-18
Degree $2$
Conductor $196$
Sign $0.576 + 0.817i$
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  + (0.846 − 1.13i)2-s + (0.635 + 0.797i)3-s + (−0.567 − 1.91i)4-s + (−0.726 + 0.579i)5-s + (1.44 − 0.0455i)6-s + (2.60 − 0.472i)7-s + (−2.65 − 0.980i)8-s + (0.436 − 1.91i)9-s + (0.0415 + 1.31i)10-s + (5.04 − 1.15i)11-s + (1.16 − 1.67i)12-s + (−2.71 + 0.618i)13-s + (1.66 − 3.34i)14-s + (−0.923 − 0.210i)15-s + (−3.35 + 2.17i)16-s + (−2.67 + 5.55i)17-s + ⋯
L(s)  = 1  + (0.598 − 0.801i)2-s + (0.367 + 0.460i)3-s + (−0.283 − 0.958i)4-s + (−0.324 + 0.259i)5-s + (0.588 − 0.0185i)6-s + (0.983 − 0.178i)7-s + (−0.937 − 0.346i)8-s + (0.145 − 0.637i)9-s + (0.0131 + 0.415i)10-s + (1.52 − 0.347i)11-s + (0.337 − 0.482i)12-s + (−0.751 + 0.171i)13-s + (0.445 − 0.895i)14-s + (−0.238 − 0.0544i)15-s + (−0.839 + 0.544i)16-s + (−0.648 + 1.34i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.576 + 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.576 + 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.576 + 0.817i$
Analytic conductor: \(1.56506\)
Root analytic conductor: \(1.25102\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :1/2),\ 0.576 + 0.817i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54232 - 0.799358i\)
\(L(\frac12)\) \(\approx\) \(1.54232 - 0.799358i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.846 + 1.13i)T \)
7 \( 1 + (-2.60 + 0.472i)T \)
good3 \( 1 + (-0.635 - 0.797i)T + (-0.667 + 2.92i)T^{2} \)
5 \( 1 + (0.726 - 0.579i)T + (1.11 - 4.87i)T^{2} \)
11 \( 1 + (-5.04 + 1.15i)T + (9.91 - 4.77i)T^{2} \)
13 \( 1 + (2.71 - 0.618i)T + (11.7 - 5.64i)T^{2} \)
17 \( 1 + (2.67 - 5.55i)T + (-10.5 - 13.2i)T^{2} \)
19 \( 1 + 5.31T + 19T^{2} \)
23 \( 1 + (-2.67 - 5.54i)T + (-14.3 + 17.9i)T^{2} \)
29 \( 1 + (6.99 + 3.37i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + 2.80T + 31T^{2} \)
37 \( 1 + (5.03 + 2.42i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (-8.64 + 6.89i)T + (9.12 - 39.9i)T^{2} \)
43 \( 1 + (1.76 + 1.40i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (-1.31 - 5.75i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-3.53 + 1.70i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + (-3.63 + 4.55i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (2.83 - 5.89i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 - 7.05iT - 67T^{2} \)
71 \( 1 + (-0.607 - 1.26i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (4.23 + 0.966i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 + 1.47iT - 79T^{2} \)
83 \( 1 + (-2.36 + 10.3i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-8.05 - 1.83i)T + (80.1 + 38.6i)T^{2} \)
97 \( 1 + 11.4iT - 97T^{2} \)
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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

−12.20443655964007949230473200448, −11.35797075459324944568552559663, −10.69519703535455199803634318293, −9.389955223287259114032485734001, −8.768485943450761848970324210022, −7.09357114992357455551968840252, −5.81169388962977913837357046484, −4.20753451333743895641702717382, −3.74902822572673252686977171572, −1.78830585698408429409879593902, 2.31381227817840939686441650973, 4.28702565813818713758929106904, 5.00188081947156964901612339771, 6.65703701427684333005423272850, 7.43215323702893397760030703346, 8.430215747507871039332508850027, 9.176606110548302843275451031845, 11.01415079559243791452776981793, 12.00463185335235171244770109469, 12.71326116579253336890074977013

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