Properties

Label 2-14e2-7.4-c7-0-15
Degree $2$
Conductor $196$
Sign $-0.386 + 0.922i$
Analytic cond. $61.2274$
Root an. cond. $7.82479$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−26.2 + 45.3i)3-s + (−99.6 − 172. i)5-s + (−279. − 484. i)9-s + (−609. + 1.05e3i)11-s + 6.44e3·13-s + 1.04e4·15-s + (−1.33e4 + 2.31e4i)17-s + (−7.45e3 − 1.29e4i)19-s + (3.67e4 + 6.36e4i)23-s + (1.92e4 − 3.32e4i)25-s − 8.52e4·27-s − 1.06e5·29-s + (−3.34e4 + 5.78e4i)31-s + (−3.19e4 − 5.53e4i)33-s + (2.39e5 + 4.15e5i)37-s + ⋯
L(s)  = 1  + (−0.560 + 0.970i)3-s + (−0.356 − 0.617i)5-s + (−0.127 − 0.221i)9-s + (−0.138 + 0.239i)11-s + 0.814·13-s + 0.798·15-s + (−0.661 + 1.14i)17-s + (−0.249 − 0.431i)19-s + (0.629 + 1.09i)23-s + (0.246 − 0.426i)25-s − 0.834·27-s − 0.810·29-s + (−0.201 + 0.348i)31-s + (−0.154 − 0.267i)33-s + (0.778 + 1.34i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $-0.386 + 0.922i$
Analytic conductor: \(61.2274\)
Root analytic conductor: \(7.82479\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :7/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.1233907454\)
\(L(\frac12)\) \(\approx\) \(0.1233907454\)
\(L(\frac{9}{2})\) 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 \( 1 + (26.2 - 45.3i)T + (-1.09e3 - 1.89e3i)T^{2} \)
5 \( 1 + (99.6 + 172. i)T + (-3.90e4 + 6.76e4i)T^{2} \)
11 \( 1 + (609. - 1.05e3i)T + (-9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 - 6.44e3T + 6.27e7T^{2} \)
17 \( 1 + (1.33e4 - 2.31e4i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (7.45e3 + 1.29e4i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + (-3.67e4 - 6.36e4i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 + 1.06e5T + 1.72e10T^{2} \)
31 \( 1 + (3.34e4 - 5.78e4i)T + (-1.37e10 - 2.38e10i)T^{2} \)
37 \( 1 + (-2.39e5 - 4.15e5i)T + (-4.74e10 + 8.22e10i)T^{2} \)
41 \( 1 + 6.44e5T + 1.94e11T^{2} \)
43 \( 1 - 1.45e5T + 2.71e11T^{2} \)
47 \( 1 + (5.05e5 + 8.75e5i)T + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 + (1.71e4 - 2.96e4i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (-2.21e5 + 3.83e5i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (-5.70e5 - 9.87e5i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (-2.15e6 + 3.74e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 - 2.54e6T + 9.09e12T^{2} \)
73 \( 1 + (-1.83e6 + 3.18e6i)T + (-5.52e12 - 9.56e12i)T^{2} \)
79 \( 1 + (4.27e6 + 7.40e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + 1.79e6T + 2.71e13T^{2} \)
89 \( 1 + (2.78e6 + 4.81e6i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 + 1.72e7T + 8.07e13T^{2} \)
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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

−10.92583458591536093198052998292, −10.03898789114404177613881807107, −8.969947741269293092522390473441, −8.079929649069061966367007538786, −6.61064088078392944903217079373, −5.37504250134008535731227265315, −4.51914883626294864979106170958, −3.56223904798319913416939771650, −1.62113719499228249659975478702, −0.03769574961545899779633908542, 1.08345390428256736115630164645, 2.52949531173661521046451577216, 3.90616793167776405555196925372, 5.47860384921091361982561338109, 6.59345680504772517635660239053, 7.18306516783960989989576724160, 8.323767138146035863083960902712, 9.537114904490032922706764408022, 11.07294294818559203321474041388, 11.28214006784498022783142891386

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