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Properties

Label 2-14e2-7.4-c3-0-9
Degree $2$
Conductor $196$
Sign $-0.991 + 0.126i$
Analytic cond. $11.5643$
Root an. cond. $3.40064$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.04 − 5.26i)3-s + (−9.58 − 16.5i)5-s + (−5 − 8.66i)9-s + (13.2 − 23.0i)11-s − 10.3·13-s − 116.·15-s + (−50.6 + 87.7i)17-s + (−46.8 − 81.1i)19-s + (4.28 + 7.42i)23-s + (−121. + 209. i)25-s + 103.·27-s + 52.3·29-s + (−27.7 + 47.9i)31-s + (−80.8 − 140. i)33-s + (−214. − 371. i)37-s + ⋯
L(s)  = 1  + (0.585 − 1.01i)3-s + (−0.857 − 1.48i)5-s + (−0.185 − 0.320i)9-s + (0.364 − 0.630i)11-s − 0.220·13-s − 2.00·15-s + (−0.722 + 1.25i)17-s + (−0.566 − 0.980i)19-s + (0.0388 + 0.0673i)23-s + (−0.969 + 1.67i)25-s + 0.737·27-s + 0.335·29-s + (−0.160 + 0.278i)31-s + (−0.426 − 0.738i)33-s + (−0.951 − 1.64i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(11.5643\)
Root analytic conductor: \(3.40064\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :3/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0841014 - 1.32526i\)
\(L(\frac12)\) \(\approx\) \(0.0841014 - 1.32526i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-3.04 + 5.26i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (9.58 + 16.5i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-13.2 + 23.0i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 10.3T + 2.19e3T^{2} \)
17 \( 1 + (50.6 - 87.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (46.8 + 81.1i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-4.28 - 7.42i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 52.3T + 2.43e4T^{2} \)
31 \( 1 + (27.7 - 47.9i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (214. + 371. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 137.T + 6.89e4T^{2} \)
43 \( 1 + 172T + 7.95e4T^{2} \)
47 \( 1 + (24.6 + 42.6i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-237. + 410. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-98.5 + 170. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (200. + 347. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (62.7 - 108. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 788.T + 3.57e5T^{2} \)
73 \( 1 + (-302. + 523. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (391. + 678. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 339.T + 5.71e5T^{2} \)
89 \( 1 + (-255. - 443. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 672.T + 9.12e5T^{2} \)
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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

−11.88029847280773071799838801912, −10.79138802123928277855956213700, −8.998483877564591449168001447467, −8.554584837587603875343612247253, −7.70200789183751556920023823076, −6.54804759597979250091830446839, −4.99180013383635942101988857785, −3.79842680354324784721103833044, −1.91581407004642967892325115095, −0.53431010427785656541803549222, 2.64256447565118468292692028749, 3.67666670266718767960509032568, 4.61284394106216474085501005039, 6.54333274822389427129270429305, 7.37765048366936998595917940608, 8.582114420886776112576849923702, 9.762733666493983781744629246069, 10.41017752248029105218107090781, 11.39189742909800507407235907304, 12.24770254019677880506097192400

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