Properties

Degree $2$
Conductor $196$
Sign $0.553 - 0.832i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 − 0.366i)2-s + (−0.866 + 1.5i)3-s + (1.73 + i)4-s + (1.5 − 0.866i)5-s + (1.73 − 1.73i)6-s + (−1.99 − 2i)8-s + (−2.36 + 0.633i)10-s + (0.866 + 0.5i)11-s + (−3 + 1.73i)12-s + 3.46i·13-s + 3i·15-s + (1.99 + 3.46i)16-s + (1.5 + 0.866i)17-s + (2.59 + 4.5i)19-s + 3.46·20-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (−0.499 + 0.866i)3-s + (0.866 + 0.5i)4-s + (0.670 − 0.387i)5-s + (0.707 − 0.707i)6-s + (−0.707 − 0.707i)8-s + (−0.748 + 0.200i)10-s + (0.261 + 0.150i)11-s + (−0.866 + 0.499i)12-s + 0.960i·13-s + 0.774i·15-s + (0.499 + 0.866i)16-s + (0.363 + 0.210i)17-s + (0.596 + 1.03i)19-s + 0.774·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.553 - 0.832i$
Motivic weight: \(1\)
Character: $\chi_{196} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :1/2),\ 0.553 - 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.664758 + 0.356207i\)
\(L(\frac12)\) \(\approx\) \(0.664758 + 0.356207i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.366i)T \)
7 \( 1 \)
good3 \( 1 + (0.866 - 1.5i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + (-1.5 - 0.866i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.59 - 4.5i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (-0.866 + 1.5i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.46iT - 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 + (4.33 + 7.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.59 + 4.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.5 + 2.59i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.59 - 1.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 14iT - 71T^{2} \)
73 \( 1 + (7.5 + 4.33i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.79 + 4.5i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.8T + 83T^{2} \)
89 \( 1 + (13.5 - 7.79i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 17.3iT - 97T^{2} \)
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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

−12.26437460544487810809447046465, −11.47572545873715318909783608070, −10.42902174285216948904302457704, −9.768341632818636771111848038472, −9.041128515753374920510891953853, −7.77953917178740500088383962357, −6.45062265678780178289965525418, −5.26238408978096231976593731906, −3.79688584054174697644332884819, −1.78396757973552578333510768553, 1.06675956187295801218069986067, 2.77137164377651728596220986912, 5.42710271713889409029199171721, 6.38658450215870992379121208599, 7.12961729600125241607792861492, 8.191299020182837639562663996814, 9.450890603626781246457357900185, 10.26268342033429524430766599388, 11.31390292320862864114643337305, 12.14469942421512230534909703327

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