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Properties

Label 2-14e2-7.3-c8-0-15
Degree $2$
Conductor $196$
Sign $0.832 + 0.553i$
Analytic cond. $79.8462$
Root an. cond. $8.93567$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (25.9 + 15.0i)3-s + (−485. + 280. i)5-s + (−2.83e3 − 4.90e3i)9-s + (5.55e3 − 9.62e3i)11-s + 3.86e4i·13-s − 1.68e4·15-s + (9.18e4 + 5.30e4i)17-s + (−1.64e5 + 9.47e4i)19-s + (−2.31e5 − 4.01e5i)23-s + (−3.82e4 + 6.63e4i)25-s − 3.66e5i·27-s − 6.65e4·29-s + (−4.71e5 − 2.72e5i)31-s + (2.88e5 − 1.66e5i)33-s + (4.59e5 + 7.96e5i)37-s + ⋯
L(s)  = 1  + (0.320 + 0.185i)3-s + (−0.776 + 0.448i)5-s + (−0.431 − 0.747i)9-s + (0.379 − 0.657i)11-s + 1.35i·13-s − 0.332·15-s + (1.09 + 0.635i)17-s + (−1.25 + 0.727i)19-s + (−0.828 − 1.43i)23-s + (−0.0980 + 0.169i)25-s − 0.690i·27-s − 0.0941·29-s + (−0.510 − 0.294i)31-s + (0.243 − 0.140i)33-s + (0.245 + 0.425i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.832 + 0.553i$
Analytic conductor: \(79.8462\)
Root analytic conductor: \(8.93567\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :4),\ 0.832 + 0.553i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.494110332\)
\(L(\frac12)\) \(\approx\) \(1.494110332\)
\(L(5)\) 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 \( 1 + (-25.9 - 15.0i)T + (3.28e3 + 5.68e3i)T^{2} \)
5 \( 1 + (485. - 280. i)T + (1.95e5 - 3.38e5i)T^{2} \)
11 \( 1 + (-5.55e3 + 9.62e3i)T + (-1.07e8 - 1.85e8i)T^{2} \)
13 \( 1 - 3.86e4iT - 8.15e8T^{2} \)
17 \( 1 + (-9.18e4 - 5.30e4i)T + (3.48e9 + 6.04e9i)T^{2} \)
19 \( 1 + (1.64e5 - 9.47e4i)T + (8.49e9 - 1.47e10i)T^{2} \)
23 \( 1 + (2.31e5 + 4.01e5i)T + (-3.91e10 + 6.78e10i)T^{2} \)
29 \( 1 + 6.65e4T + 5.00e11T^{2} \)
31 \( 1 + (4.71e5 + 2.72e5i)T + (4.26e11 + 7.38e11i)T^{2} \)
37 \( 1 + (-4.59e5 - 7.96e5i)T + (-1.75e12 + 3.04e12i)T^{2} \)
41 \( 1 + 2.89e6iT - 7.98e12T^{2} \)
43 \( 1 - 3.05e6T + 1.16e13T^{2} \)
47 \( 1 + (-9.60e4 + 5.54e4i)T + (1.19e13 - 2.06e13i)T^{2} \)
53 \( 1 + (-6.87e6 + 1.19e7i)T + (-3.11e13 - 5.39e13i)T^{2} \)
59 \( 1 + (5.62e5 + 3.24e5i)T + (7.34e13 + 1.27e14i)T^{2} \)
61 \( 1 + (-7.64e6 + 4.41e6i)T + (9.58e13 - 1.66e14i)T^{2} \)
67 \( 1 + (-2.44e6 + 4.24e6i)T + (-2.03e14 - 3.51e14i)T^{2} \)
71 \( 1 - 1.20e7T + 6.45e14T^{2} \)
73 \( 1 + (-2.76e7 - 1.59e7i)T + (4.03e14 + 6.98e14i)T^{2} \)
79 \( 1 + (-3.40e7 - 5.89e7i)T + (-7.58e14 + 1.31e15i)T^{2} \)
83 \( 1 + 4.61e7iT - 2.25e15T^{2} \)
89 \( 1 + (9.23e6 - 5.33e6i)T + (1.96e15 - 3.40e15i)T^{2} \)
97 \( 1 + 4.77e7iT - 7.83e15T^{2} \)
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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

−11.02756147698005778321397692988, −9.935860485980403942336006863354, −8.791294325945196943953746402616, −8.105119387137996940851786867494, −6.76179820705274412260592268659, −5.93737812280839594502699261823, −4.07457843202608207510534808115, −3.60364692894497165963620878728, −2.09407887617127719659416931978, −0.44789693650413028288992961490, 0.805694839277697863477851984168, 2.27808995641941017519336030669, 3.52526898106326438089059073375, 4.75124812868999115007259734033, 5.80257271079907861971044087903, 7.47193529748851650971732251219, 7.920789784426680109855599691801, 8.985486831254476857776362827088, 10.14083799303887285888635573405, 11.18480563801457132235555651293

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