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Properties

Label 2-14e2-196.111-c1-0-1
Degree $2$
Conductor $196$
Sign $0.219 - 0.975i$
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  + (−0.368 − 1.36i)2-s + (−1.56 − 1.95i)3-s + (−1.72 + 1.00i)4-s + (−2.80 + 2.23i)5-s + (−2.09 + 2.85i)6-s + (2.40 − 1.09i)7-s + (2.00 + 1.99i)8-s + (−0.728 + 3.19i)9-s + (4.08 + 3.00i)10-s + (−3.69 + 0.843i)11-s + (4.66 + 1.81i)12-s + (−0.397 + 0.0908i)13-s + (−2.38 − 2.88i)14-s + (8.76 + 2.00i)15-s + (1.97 − 3.47i)16-s + (−1.99 + 4.14i)17-s + ⋯
L(s)  = 1  + (−0.260 − 0.965i)2-s + (−0.901 − 1.13i)3-s + (−0.864 + 0.502i)4-s + (−1.25 + 1.00i)5-s + (−0.856 + 1.16i)6-s + (0.909 − 0.414i)7-s + (0.710 + 0.703i)8-s + (−0.242 + 1.06i)9-s + (1.29 + 0.951i)10-s + (−1.11 + 0.254i)11-s + (1.34 + 0.523i)12-s + (−0.110 + 0.0251i)13-s + (−0.637 − 0.770i)14-s + (2.26 + 0.516i)15-s + (0.494 − 0.869i)16-s + (−0.483 + 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.219 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.219 - 0.975i$
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: \((2,\ 196,\ (\ :1/2),\ 0.219 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0545740 + 0.0436740i\)
\(L(\frac12)\) \(\approx\) \(0.0545740 + 0.0436740i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.368 + 1.36i)T \)
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 \( 1 + 1.47iT - 97T^{2} \)
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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

−12.50374430923350875200872568738, −11.56101485183802465325166326425, −10.96937456442211891043464037255, −10.45176754514252179396541752792, −8.374959938551320844948381440305, −7.66694895980453917500118517843, −6.86174424459537540153358739944, −5.10667739076210688426671499310, −3.72775860130623268302482811147, −1.97508749739974956318094550168, 0.07356055962894261887860081863, 4.18026481827591657747882959626, 4.93646849580258050938892439579, 5.51414785263389617328117657678, 7.31292284230162527226556120079, 8.359531043849727880438867577786, 9.003015287452066897193022226692, 10.37744765769235518016264614320, 11.18338301481807514329531773808, 12.11925523179493761012302212454

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