↑

Properties

Label 2-14e2-196.111-c1-0-3
Degree $2$
Conductor $196$
Sign $0.368 - 0.929i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.260 − 1.38i)2-s + (1.56 + 1.95i)3-s + (−1.86 + 0.724i)4-s + (−2.80 + 2.23i)5-s + (2.31 − 2.68i)6-s + (−2.40 + 1.09i)7-s + (1.49 + 2.40i)8-s + (−0.728 + 3.19i)9-s + (3.84 + 3.31i)10-s + (3.69 − 0.843i)11-s + (−4.33 − 2.51i)12-s + (−0.397 + 0.0908i)13-s + (2.15 + 3.06i)14-s + (−8.76 − 2.00i)15-s + (2.94 − 2.70i)16-s + (−1.99 + 4.14i)17-s + ⋯
L(s)  = 1  + (−0.184 − 0.982i)2-s + (0.901 + 1.13i)3-s + (−0.932 + 0.362i)4-s + (−1.25 + 1.00i)5-s + (0.945 − 1.09i)6-s + (−0.909 + 0.414i)7-s + (0.527 + 0.849i)8-s + (−0.242 + 1.06i)9-s + (1.21 + 1.04i)10-s + (1.11 − 0.254i)11-s + (−1.25 − 0.727i)12-s + (−0.110 + 0.0251i)13-s + (0.575 + 0.817i)14-s + (−2.26 − 0.516i)15-s + (0.737 − 0.675i)16-s + (−0.483 + 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.368 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.368 - 0.929i$
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: \((2,\ 196,\ (\ :1/2),\ 0.368 - 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.786301 + 0.534369i\)
\(L(\frac12)\) \(\approx\) \(0.786301 + 0.534369i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.260 + 1.38i)T \)
7 \( 1 + (2.40 - 1.09i)T \)
good3 \( 1 + (-1.56 - 1.95i)T + (-0.667 + 2.92i)T^{2} \)
5 \( 1 + (2.80 - 2.23i)T + (1.11 - 4.87i)T^{2} \)
11 \( 1 + (-3.69 + 0.843i)T + (9.91 - 4.77i)T^{2} \)
13 \( 1 + (0.397 - 0.0908i)T + (11.7 - 5.64i)T^{2} \)
17 \( 1 + (1.99 - 4.14i)T + (-10.5 - 13.2i)T^{2} \)
19 \( 1 - 3.39T + 19T^{2} \)
23 \( 1 + (0.764 + 1.58i)T + (-14.3 + 17.9i)T^{2} \)
29 \( 1 + (-3.44 - 1.65i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 - 10.4T + 31T^{2} \)
37 \( 1 + (4.97 + 2.39i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (6.23 - 4.97i)T + (9.12 - 39.9i)T^{2} \)
43 \( 1 + (1.08 + 0.861i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (-1.15 - 5.04i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (1.01 - 0.489i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + (6.97 - 8.74i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (-3.88 + 8.07i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 + 5.92iT - 67T^{2} \)
71 \( 1 + (-6.36 - 13.2i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (-2.53 - 0.577i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 + 5.63iT - 79T^{2} \)
83 \( 1 + (-1.68 + 7.39i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (1.80 + 0.412i)T + (80.1 + 38.6i)T^{2} \)
97 \( 1 + 1.47iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.33110116735022137743660423492, −11.61700543632254417182121817901, −10.57837202126458997155074767396, −9.865521559178069786668557736951, −8.893030854334509166708810977929, −8.143131623813210002118204105247, −6.59496377346082213749968979348, −4.40506144275526903466094197103, −3.55527678373393833795133491740, −2.91245064194231281319080962630, 0.874593366708727650838699863582, 3.52717631963846902545325852844, 4.74541141022490771529566888432, 6.62655763930438922356575774861, 7.23531551428699113796381133794, 8.159893593026767152306923384840, 8.890177985555829950334000886197, 9.786970077325989620924653445007, 11.87476939412819386622768921370, 12.44870289435516281832357762296

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