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Properties

Label 2-98-7.2-c3-0-8
Degree $2$
Conductor $98$
Sign $-0.991 - 0.126i$
Analytic cond. $5.78218$
Root an. cond. $2.40461$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)2-s + (−4.69 − 8.12i)3-s + (−1.99 − 3.46i)4-s + (4.69 − 8.12i)5-s + 18.7·6-s + 7.99·8-s + (−30.5 + 52.8i)9-s + (9.38 + 16.2i)10-s + (−10 − 17.3i)11-s + (−18.7 + 32.4i)12-s − 65.6·13-s − 88.0·15-s + (−8 + 13.8i)16-s + (28.1 + 48.7i)17-s + (−61 − 105. i)18-s + (4.69 − 8.12i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.902 − 1.56i)3-s + (−0.249 − 0.433i)4-s + (0.419 − 0.726i)5-s + 1.27·6-s + 0.353·8-s + (−1.12 + 1.95i)9-s + (0.296 + 0.513i)10-s + (−0.274 − 0.474i)11-s + (−0.451 + 0.781i)12-s − 1.40·13-s − 1.51·15-s + (−0.125 + 0.216i)16-s + (0.401 + 0.695i)17-s + (−0.798 − 1.38i)18-s + (0.0566 − 0.0980i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $-0.991 - 0.126i$
Analytic conductor: \(5.78218\)
Root analytic conductor: \(2.40461\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :3/2),\ -0.991 - 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0229422 + 0.361522i\)
\(L(\frac12)\) \(\approx\) \(0.0229422 + 0.361522i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - 1.73i)T \)
7 \( 1 \)
good3 \( 1 + (4.69 + 8.12i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-4.69 + 8.12i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (10 + 17.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 65.6T + 2.19e3T^{2} \)
17 \( 1 + (-28.1 - 48.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-4.69 + 8.12i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (24 - 41.5i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 166T + 2.43e4T^{2} \)
31 \( 1 + (103. + 178. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-39 + 67.5i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 393.T + 6.89e4T^{2} \)
43 \( 1 - 436T + 7.95e4T^{2} \)
47 \( 1 + (-103. + 178. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (31 + 53.6i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (333. + 576. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-136. + 235. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (290 + 502. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 544T + 3.57e5T^{2} \)
73 \( 1 + (300. + 519. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-340 + 588. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 196.T + 5.71e5T^{2} \)
89 \( 1 + (750. - 1.29e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 656.T + 9.12e5T^{2} \)
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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

−12.89713230748718309861987177058, −12.08708230803099149625628226931, −10.87130390184026256107190888922, −9.409183659009238899853363886952, −8.008549585650892827207378958293, −7.23496980258033751371402825953, −5.94545674548936142305268069061, −5.20012393899947765260331933286, −1.81350884353071380397138412146, −0.25101696356901132321400637820, 2.85843559049178382146361478411, 4.43314697378421512934346563288, 5.53957525570828792871722273548, 7.18310907878607353881183457087, 9.146562065883587131643315769404, 10.04176033724342837725100015557, 10.48890439157576424321597970986, 11.57906667667770828787321147779, 12.45752750190181179968033367445, 14.25355213940592940117272383602

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