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Properties

Label 2-14e2-28.27-c1-0-9
Degree $2$
Conductor $196$
Sign $0.436 + 0.899i$
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  + (0.207 + 1.39i)2-s − 2.14·3-s + (−1.91 + 0.579i)4-s − 2.61i·5-s + (−0.443 − 2.99i)6-s + (−1.20 − 2.55i)8-s + 1.58·9-s + (3.65 − 0.541i)10-s − 3.95i·11-s + (4.09 − 1.24i)12-s − 1.08i·13-s + 5.59i·15-s + (3.32 − 2.21i)16-s − 0.317i·17-s + (0.328 + 2.21i)18-s − 5.16·19-s + ⋯
L(s)  = 1  + (0.146 + 0.989i)2-s − 1.23·3-s + (−0.957 + 0.289i)4-s − 1.16i·5-s + (−0.181 − 1.22i)6-s + (−0.426 − 0.904i)8-s + 0.528·9-s + (1.15 − 0.171i)10-s − 1.19i·11-s + (1.18 − 0.358i)12-s − 0.300i·13-s + 1.44i·15-s + (0.832 − 0.554i)16-s − 0.0768i·17-s + (0.0774 + 0.522i)18-s − 1.18·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.436 + 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.436 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.436 + 0.899i$
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: \((2,\ 196,\ (\ :1/2),\ 0.436 + 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.409324 - 0.256523i\)
\(L(\frac12)\) \(\approx\) \(0.409324 - 0.256523i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.207 - 1.39i)T \)
7 \( 1 \)
good3 \( 1 + 2.14T + 3T^{2} \)
5 \( 1 + 2.61iT - 5T^{2} \)
11 \( 1 + 3.95iT - 11T^{2} \)
13 \( 1 + 1.08iT - 13T^{2} \)
17 \( 1 + 0.317iT - 17T^{2} \)
19 \( 1 + 5.16T + 19T^{2} \)
23 \( 1 + 2.31iT - 23T^{2} \)
29 \( 1 + 6.82T + 29T^{2} \)
31 \( 1 + 6.05T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 - 2.29iT - 41T^{2} \)
43 \( 1 + 7.23iT - 43T^{2} \)
47 \( 1 - 4.28T + 47T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 - 11.2T + 59T^{2} \)
61 \( 1 + 5.41iT - 61T^{2} \)
67 \( 1 - 3.27iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 14.0iT - 73T^{2} \)
79 \( 1 - 7.91iT - 79T^{2} \)
83 \( 1 - 9.45T + 83T^{2} \)
89 \( 1 + 5.99iT - 89T^{2} \)
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

−12.58505827028343965682371694521, −11.48415404769297728368676892704, −10.42603067864684207266518608166, −8.953122588872202444404836533612, −8.379251174350100145361932492097, −6.91934734216391750707782101772, −5.75026478226227269792671003676, −5.28811315835543709878953368168, −4.00942158098649796032886784373, −0.46716752283052425147208377624, 2.15618108878931084630583439610, 3.85410677659243061116880662377, 5.12896278546064518009199117064, 6.26941110403255502457797301847, 7.34671487538195709015513592971, 9.089306113882380508818890178081, 10.27112901194895225619652832004, 10.79292837268589364993161105495, 11.57997915610970262007731542378, 12.37783379207564952098091534340

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