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Properties

Label 2-98-7.6-c12-0-8
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 − 1.13e3i·3-s + 2.04e3·4-s − 1.40e4i·5-s + 5.15e4i·6-s − 9.26e4·8-s − 7.63e5·9-s + 6.34e5i·10-s − 3.04e6·11-s − 2.33e6i·12-s + 8.06e6i·13-s − 1.59e7·15-s + 4.19e6·16-s + 3.37e6i·17-s + 3.45e7·18-s − 1.59e7i·19-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.56i·3-s + 0.500·4-s − 0.896i·5-s + 1.10i·6-s − 0.353·8-s − 1.43·9-s + 0.634i·10-s − 1.71·11-s − 0.780i·12-s + 1.67i·13-s − 1.39·15-s + 0.250·16-s + 0.139i·17-s + 1.01·18-s − 0.338i·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

\(L(\frac{13}{2})\) \(\approx\) \(0.8256733435\)
\(L(\frac12)\) \(\approx\) \(0.8256733435\)
\(L(7)\) 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 \( 1 + 1.13e3iT - 5.31e5T^{2} \)
5 \( 1 + 1.40e4iT - 2.44e8T^{2} \)
11 \( 1 + 3.04e6T + 3.13e12T^{2} \)
13 \( 1 - 8.06e6iT - 2.32e13T^{2} \)
17 \( 1 - 3.37e6iT - 5.82e14T^{2} \)
19 \( 1 + 1.59e7iT - 2.21e15T^{2} \)
23 \( 1 + 1.21e8T + 2.19e16T^{2} \)
29 \( 1 - 1.10e9T + 3.53e17T^{2} \)
31 \( 1 - 1.72e8iT - 7.87e17T^{2} \)
37 \( 1 + 1.66e9T + 6.58e18T^{2} \)
41 \( 1 - 2.84e9iT - 2.25e19T^{2} \)
43 \( 1 - 3.52e9T + 3.99e19T^{2} \)
47 \( 1 + 5.48e9iT - 1.16e20T^{2} \)
53 \( 1 + 7.09e9T + 4.91e20T^{2} \)
59 \( 1 + 3.39e10iT - 1.77e21T^{2} \)
61 \( 1 - 7.07e10iT - 2.65e21T^{2} \)
67 \( 1 + 1.15e11T + 8.18e21T^{2} \)
71 \( 1 + 5.03e10T + 1.64e22T^{2} \)
73 \( 1 - 7.59e10iT - 2.29e22T^{2} \)
79 \( 1 - 2.57e11T + 5.90e22T^{2} \)
83 \( 1 + 2.08e9iT - 1.06e23T^{2} \)
89 \( 1 + 8.72e11iT - 2.46e23T^{2} \)
97 \( 1 - 8.85e11iT - 6.93e23T^{2} \)
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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

−11.67592648431165794242399152266, −10.31080319114419493468080858035, −8.875164117486006098431378341634, −8.144410816807145880962286254219, −7.19913416464560509441862006988, −6.20228996390391561432485974345, −4.79438213683155223434954506666, −2.59284954536306539436034919798, −1.68861594463770368659191075017, −0.68824121295385475740019404894, 0.34714814153091060811850782903, 2.63175412490890668588273072890, 3.28464002159045907737064389391, 4.89803507208967457060767123246, 5.92281386216569137123743769238, 7.59626769959108393321264453477, 8.487448133781994050510583444415, 9.972180909228804954693149357749, 10.41179292171602558769405874359, 10.91617741200828934176535924246

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