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Properties

Label 2-14e2-28.27-c1-0-8
Degree $2$
Conductor $196$
Sign $0.780 + 0.624i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.780 + 0.624i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.780 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.780 + 0.624i$
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: \((2,\ 196,\ (\ :1/2),\ 0.780 + 0.624i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21802 - 0.427152i\)
\(L(\frac12)\) \(\approx\) \(1.21802 - 0.427152i\)
\(L(\frac{3}{2})\) 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

−12.60569026108998688109391045578, −11.22130653941108820955842352869, −10.20151309460662366549188221162, −9.086178000638015554242354364875, −8.636510138606405443659294958255, −7.85451851550882707706896096075, −6.57024297313048804462860529552, −4.25240455574224637362817880871, −3.10815144370129955268898521970, −1.77342459507973762169906803848, 2.06612843312065820857915354815, 3.36512438170544929109558994611, 5.29370109926555080772497946568, 7.04476476092919752224322518513, 7.54776010073388849019245105259, 8.635179292770443742854602596786, 9.424775700756348959622140041145, 10.19762929038482261768361222172, 11.33976637955894367173259094571, 12.85554261334316762134359117647

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