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Properties

Label 2-98-7.2-c5-0-7
Degree $2$
Conductor $98$
Sign $0.266 - 0.963i$
Analytic cond. $15.7176$
Root an. cond. $3.96454$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 + 3.46i)2-s + (4 + 6.92i)3-s + (−7.99 − 13.8i)4-s + (5 − 8.66i)5-s − 31.9·6-s + 63.9·8-s + (89.5 − 155. i)9-s + (20 + 34.6i)10-s + (170 + 294. i)11-s + (63.9 − 110. i)12-s + 294·13-s + 80·15-s + (−128 + 221. i)16-s + (613 + 1.06e3i)17-s + (358 + 620. i)18-s + (1.21e3 − 2.10e3i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.256 + 0.444i)3-s + (−0.249 − 0.433i)4-s + (0.0894 − 0.154i)5-s − 0.362·6-s + 0.353·8-s + (0.368 − 0.637i)9-s + (0.0632 + 0.109i)10-s + (0.423 + 0.733i)11-s + (0.128 − 0.222i)12-s + 0.482·13-s + 0.0918·15-s + (−0.125 + 0.216i)16-s + (0.514 + 0.891i)17-s + (0.260 + 0.451i)18-s + (0.772 − 1.33i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.266 - 0.963i$
Analytic conductor: \(15.7176\)
Root analytic conductor: \(3.96454\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :5/2),\ 0.266 - 0.963i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.40469 + 1.06863i\)
\(L(\frac12)\) \(\approx\) \(1.40469 + 1.06863i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2 - 3.46i)T \)
7 \( 1 \)
good3 \( 1 + (-4 - 6.92i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (-5 + 8.66i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-170 - 294. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 294T + 3.71e5T^{2} \)
17 \( 1 + (-613 - 1.06e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-1.21e3 + 2.10e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (1.00e3 - 1.73e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 6.74e3T + 2.05e7T^{2} \)
31 \( 1 + (-4.42e3 - 7.66e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (4.59e3 - 7.95e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 1.45e4T + 1.15e8T^{2} \)
43 \( 1 - 8.10e3T + 1.47e8T^{2} \)
47 \( 1 + (156 - 270. i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-7.31e3 - 1.26e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (1.38e4 + 2.39e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-1.71e4 + 2.97e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (6.15e3 + 1.06e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 3.69e4T + 1.80e9T^{2} \)
73 \( 1 + (3.08e4 + 5.34e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-3.23e4 + 5.60e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 7.70e4T + 3.93e9T^{2} \)
89 \( 1 + (4.08e3 - 7.07e3i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 2.06e4T + 8.58e9T^{2} \)
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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

−13.33167229204887440184165386781, −12.18977843136608508826676517796, −10.79564491667310531605785523060, −9.580235374062774738262237550961, −9.004362313968316890369180351948, −7.56544129747300337121846496260, −6.45343580312168920856036236687, −4.99533068966914636599288814763, −3.59433693640265753009694177848, −1.26790760072664839952306218185, 0.951773173265855325772957131715, 2.44248386899417098885041370125, 3.95722977520276445328771021313, 5.79486229727359108365018410088, 7.39097574345423083230729532353, 8.302461948368666604670333827220, 9.556760787207023670902264270398, 10.61289694244564348629076968031, 11.65514350389488905911826455548, 12.67512342289275162792482786010

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