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Properties

Label 2-14e2-28.27-c1-0-4
Degree $2$
Conductor $196$
Sign $0.0532 + 0.998i$
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  + (−1.20 − 0.736i)2-s − 2.72·3-s + (0.914 + 1.77i)4-s + 1.08i·5-s + (3.28 + 2.00i)6-s + (0.207 − 2.82i)8-s + 4.41·9-s + (0.797 − 1.30i)10-s − 2.08i·11-s + (−2.48 − 4.84i)12-s − 2.61i·13-s − 2.94i·15-s + (−2.32 + 3.25i)16-s − 4.46i·17-s + (−5.32 − 3.25i)18-s + 1.12·19-s + ⋯
L(s)  = 1  + (−0.853 − 0.521i)2-s − 1.57·3-s + (0.457 + 0.889i)4-s + 0.484i·5-s + (1.34 + 0.819i)6-s + (0.0732 − 0.997i)8-s + 1.47·9-s + (0.252 − 0.413i)10-s − 0.628i·11-s + (−0.718 − 1.39i)12-s − 0.724i·13-s − 0.760i·15-s + (−0.582 + 0.813i)16-s − 1.08i·17-s + (−1.25 − 0.766i)18-s + 0.258·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0532 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0532 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.0532 + 0.998i$
Analytic conductor: \(1.56506\)
Root analytic conductor: \(1.25102\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (195, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :1/2),\ 0.0532 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.299080 - 0.283559i\)
\(L(\frac12)\) \(\approx\) \(0.299080 - 0.283559i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.20 + 0.736i)T \)
7 \( 1 \)
good3 \( 1 + 2.72T + 3T^{2} \)
5 \( 1 - 1.08iT - 5T^{2} \)
11 \( 1 + 2.08iT - 11T^{2} \)
13 \( 1 + 2.61iT - 13T^{2} \)
17 \( 1 + 4.46iT - 17T^{2} \)
19 \( 1 - 1.12T + 19T^{2} \)
23 \( 1 + 7.11iT - 23T^{2} \)
29 \( 1 + 1.17T + 29T^{2} \)
31 \( 1 - 7.70T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + 5.54iT - 41T^{2} \)
43 \( 1 - 7.97iT - 43T^{2} \)
47 \( 1 - 5.44T + 47T^{2} \)
53 \( 1 + 6.48T + 53T^{2} \)
59 \( 1 + 8.82T + 59T^{2} \)
61 \( 1 + 13.0iT - 61T^{2} \)
67 \( 1 + 10.0iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 7.97iT - 73T^{2} \)
79 \( 1 - 4.16iT - 79T^{2} \)
83 \( 1 - 4.31T + 83T^{2} \)
89 \( 1 - 4.01iT - 89T^{2} \)
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

−12.02847854001605473438601064098, −11.00659350778283463826863749128, −10.67771479544341039227885645730, −9.623207018556396012105567392074, −8.259670513837758288802145750579, −7.01231066055436697491438964714, −6.18930442702432291461816818439, −4.79266908020305385633819642237, −2.96148102147519891048671828583, −0.63934439657540807471685004895, 1.40787422091397484266547450780, 4.58876240931248657532975810816, 5.58745473018321959647335972818, 6.50357400956849920375043715108, 7.46501385537374386473252946185, 8.809635164545140731279871914535, 9.888412732603840093423409462875, 10.70036102182710433555941967119, 11.65376622775875629891577951401, 12.32285411193989633106097757966

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