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Properties

Label 2-14e2-28.27-c5-0-30
Degree $2$
Conductor $196$
Sign $0.997 + 0.0737i$
Analytic cond. $31.4352$
Root an. cond. $5.60671$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.48 − 1.38i)2-s − 26.5·3-s + (28.1 + 15.2i)4-s + 52.9i·5-s + (145. + 36.7i)6-s + (−133. − 122. i)8-s + 460.·9-s + (73.4 − 290. i)10-s + 105. i·11-s + (−746. − 403. i)12-s − 827. i·13-s − 1.40e3i·15-s + (561. + 856. i)16-s + 1.75e3i·17-s + (−2.52e3 − 638. i)18-s − 3.02e3·19-s + ⋯
L(s)  = 1  + (−0.969 − 0.245i)2-s − 1.70·3-s + (0.879 + 0.475i)4-s + 0.947i·5-s + (1.64 + 0.417i)6-s + (−0.736 − 0.676i)8-s + 1.89·9-s + (0.232 − 0.919i)10-s + 0.263i·11-s + (−1.49 − 0.808i)12-s − 1.35i·13-s − 1.61i·15-s + (0.548 + 0.836i)16-s + 1.47i·17-s + (−1.83 − 0.464i)18-s − 1.92·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0737i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.997 + 0.0737i$
Analytic conductor: \(31.4352\)
Root analytic conductor: \(5.60671\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (195, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :5/2),\ 0.997 + 0.0737i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.3921898852\)
\(L(\frac12)\) \(\approx\) \(0.3921898852\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.48 + 1.38i)T \)
7 \( 1 \)
good3 \( 1 + 26.5T + 243T^{2} \)
5 \( 1 - 52.9iT - 3.12e3T^{2} \)
11 \( 1 - 105. iT - 1.61e5T^{2} \)
13 \( 1 + 827. iT - 3.71e5T^{2} \)
17 \( 1 - 1.75e3iT - 1.41e6T^{2} \)
19 \( 1 + 3.02e3T + 2.47e6T^{2} \)
23 \( 1 + 2.13e3iT - 6.43e6T^{2} \)
29 \( 1 + 3.80e3T + 2.05e7T^{2} \)
31 \( 1 + 7.10e3T + 2.86e7T^{2} \)
37 \( 1 + 202.T + 6.93e7T^{2} \)
41 \( 1 + 7.21e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.39e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.33e4T + 2.29e8T^{2} \)
53 \( 1 - 1.19e4T + 4.18e8T^{2} \)
59 \( 1 - 1.30e3T + 7.14e8T^{2} \)
61 \( 1 - 1.35e4iT - 8.44e8T^{2} \)
67 \( 1 + 3.99e4iT - 1.35e9T^{2} \)
71 \( 1 + 2.95e4iT - 1.80e9T^{2} \)
73 \( 1 - 5.43e4iT - 2.07e9T^{2} \)
79 \( 1 - 8.69e4iT - 3.07e9T^{2} \)
83 \( 1 - 5.07e4T + 3.93e9T^{2} \)
89 \( 1 + 2.16e4iT - 5.58e9T^{2} \)
97 \( 1 - 4.60e3iT - 8.58e9T^{2} \)
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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

−11.10680642571416892728629592340, −10.60959335819753039035655333236, −10.29261188527787930677842400323, −8.602561825466709710017050985228, −7.32878637896082123247405620239, −6.46233979988778082498588790008, −5.68478279950016108007108939676, −3.86929578838428002571764211541, −2.07856697561917344190642678616, −0.43126290116962494598141457186, 0.53512234954362746254414966515, 1.75047927645228496508674710433, 4.49744371049738628762846180652, 5.48907396848420525809489886772, 6.46250955907779138377406378807, 7.34918221088904000528831225102, 8.854189234321673167395621106587, 9.568450919977591254188027637982, 10.83315790401838923868671559591, 11.43449405017763717233283341832

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