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Properties

Label 2-14e2-28.3-c1-0-12
Degree $2$
Conductor $196$
Sign $0.552 + 0.833i$
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  + (−1.31 + 0.520i)2-s + (1.07 − 1.85i)3-s + (1.45 − 1.36i)4-s + (2.26 − 1.30i)5-s + (−0.443 + 2.99i)6-s + (−1.20 + 2.55i)8-s + (−0.792 − 1.37i)9-s + (−2.29 + 2.89i)10-s + (−3.42 − 1.97i)11-s + (−0.974 − 4.17i)12-s + 1.08i·13-s − 5.59i·15-s + (0.257 − 3.99i)16-s + (−0.274 − 0.158i)17-s + (1.75 + 1.39i)18-s + (2.58 + 4.47i)19-s + ⋯
L(s)  = 1  + (−0.929 + 0.367i)2-s + (0.618 − 1.07i)3-s + (0.729 − 0.684i)4-s + (1.01 − 0.584i)5-s + (−0.181 + 1.22i)6-s + (−0.426 + 0.904i)8-s + (−0.264 − 0.457i)9-s + (−0.726 + 0.915i)10-s + (−1.03 − 0.596i)11-s + (−0.281 − 1.20i)12-s + 0.300i·13-s − 1.44i·15-s + (0.0642 − 0.997i)16-s + (−0.0665 − 0.0384i)17-s + (0.414 + 0.328i)18-s + (0.593 + 1.02i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.552 + 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.552 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.552 + 0.833i$
Analytic conductor: \(1.56506\)
Root analytic conductor: \(1.25102\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :1/2),\ 0.552 + 0.833i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.945862 - 0.508160i\)
\(L(\frac12)\) \(\approx\) \(0.945862 - 0.508160i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.31 - 0.520i)T \)
7 \( 1 \)
good3 \( 1 + (-1.07 + 1.85i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.26 + 1.30i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.42 + 1.97i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.08iT - 13T^{2} \)
17 \( 1 + (0.274 + 0.158i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.58 - 4.47i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.00 + 1.15i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.82T + 29T^{2} \)
31 \( 1 + (-3.02 + 5.24i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.29iT - 41T^{2} \)
43 \( 1 - 7.23iT - 43T^{2} \)
47 \( 1 + (2.14 + 3.70i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.24 - 9.08i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.61 - 9.72i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.68 + 2.70i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.83 - 1.63i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-12.1 - 7.00i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.85 - 3.95i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.45T + 83T^{2} \)
89 \( 1 + (-5.18 + 2.99i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 9.23iT - 97T^{2} \)
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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

−12.54634347141820534955676298314, −11.23631128459334776926767549746, −10.06021314043812272411900753622, −9.204743620582243867496048485085, −8.199705420066895397734705925181, −7.54718023352524609674752984942, −6.27408112223557108238123634065, −5.35536615831048946751305490123, −2.59037168517978825020408292475, −1.40257585582403092765191150702, 2.32973630888523077956747711514, 3.35767936446887287276749195036, 5.09802891377153534204900970204, 6.68694306565196463612679616911, 7.83675977891354646367482483902, 9.101001357007866987166384795740, 9.708988350736793145164331049049, 10.40270087756830114901014617815, 11.13566305943150436075268985230, 12.63552837026570441365823185831

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