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Properties

Label 2-98-7.6-c12-0-7
Degree $2$
Conductor $98$
Sign $-0.755 + 0.654i$
Analytic cond. $89.5713$
Root an. cond. $9.46421$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 45.2·2-s + 811. i·3-s + 2.04e3·4-s + 2.11e4i·5-s + 3.67e4i·6-s + 9.26e4·8-s − 1.26e5·9-s + 9.56e5i·10-s − 1.30e6·11-s + 1.66e6i·12-s + 7.09e6i·13-s − 1.71e7·15-s + 4.19e6·16-s − 3.22e7i·17-s − 5.73e6·18-s + 2.68e7i·19-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.11i·3-s + 0.500·4-s + 1.35i·5-s + 0.786i·6-s + 0.353·8-s − 0.238·9-s + 0.956i·10-s − 0.735·11-s + 0.556i·12-s + 1.47i·13-s − 1.50·15-s + 0.250·16-s − 1.33i·17-s − 0.168·18-s + 0.570i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $-0.755 + 0.654i$
Analytic conductor: \(89.5713\)
Root analytic conductor: \(9.46421\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :6),\ -0.755 + 0.654i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(2.124025552\)
\(L(\frac12)\) \(\approx\) \(2.124025552\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 45.2T \)
7 \( 1 \)
good3 \( 1 - 811. iT - 5.31e5T^{2} \)
5 \( 1 - 2.11e4iT - 2.44e8T^{2} \)
11 \( 1 + 1.30e6T + 3.13e12T^{2} \)
13 \( 1 - 7.09e6iT - 2.32e13T^{2} \)
17 \( 1 + 3.22e7iT - 5.82e14T^{2} \)
19 \( 1 - 2.68e7iT - 2.21e15T^{2} \)
23 \( 1 + 1.24e8T + 2.19e16T^{2} \)
29 \( 1 + 7.52e8T + 3.53e17T^{2} \)
31 \( 1 - 1.09e9iT - 7.87e17T^{2} \)
37 \( 1 - 3.87e9T + 6.58e18T^{2} \)
41 \( 1 + 2.55e9iT - 2.25e19T^{2} \)
43 \( 1 + 4.36e9T + 3.99e19T^{2} \)
47 \( 1 + 9.28e9iT - 1.16e20T^{2} \)
53 \( 1 - 2.18e8T + 4.91e20T^{2} \)
59 \( 1 - 6.29e10iT - 1.77e21T^{2} \)
61 \( 1 + 4.84e10iT - 2.65e21T^{2} \)
67 \( 1 - 1.09e11T + 8.18e21T^{2} \)
71 \( 1 + 7.43e10T + 1.64e22T^{2} \)
73 \( 1 + 2.61e11iT - 2.29e22T^{2} \)
79 \( 1 - 2.41e11T + 5.90e22T^{2} \)
83 \( 1 + 2.09e11iT - 1.06e23T^{2} \)
89 \( 1 + 2.06e10iT - 2.46e23T^{2} \)
97 \( 1 - 4.74e11iT - 6.93e23T^{2} \)
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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

−11.93033359668215553904309620176, −11.06034035629222751303805577860, −10.25918893505318558179309501281, −9.333048439379747171589733306441, −7.50619049015793813272589934729, −6.58210431636147243755572599462, −5.23935169704380940532261938660, −4.13628569835705448211896835723, −3.20384097442448942989508481248, −2.07442811551831966425549831462, 0.34700709033868638812335818820, 1.32148863483688236847998673646, 2.43769080667143866758241748026, 4.06851528809693051480547467421, 5.32444388266911387595496252366, 6.16202299646259096699264643217, 7.74343687828090190876393942151, 8.199834580133726904227286447803, 9.857521957318663622644281146065, 11.22907267142159045668683801675

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