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Properties

Label 2-98-49.15-c1-0-4
Degree $2$
Conductor $98$
Sign $0.430 + 0.902i$
Analytic cond. $0.782533$
Root an. cond. $0.884609$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.222 − 0.974i)2-s + (−0.234 + 0.293i)3-s + (−0.900 − 0.433i)4-s + (1.33 − 1.67i)5-s + (0.234 + 0.293i)6-s + (0.635 − 2.56i)7-s + (−0.623 + 0.781i)8-s + (0.636 + 2.78i)9-s + (−1.33 − 1.67i)10-s + (0.476 − 2.08i)11-s + (0.338 − 0.162i)12-s + (−1.27 + 5.57i)13-s + (−2.36 − 1.19i)14-s + (0.178 + 0.783i)15-s + (0.623 + 0.781i)16-s + (−4.41 + 2.12i)17-s + ⋯
L(s)  = 1  + (0.157 − 0.689i)2-s + (−0.135 + 0.169i)3-s + (−0.450 − 0.216i)4-s + (0.597 − 0.748i)5-s + (0.0955 + 0.119i)6-s + (0.240 − 0.970i)7-s + (−0.220 + 0.276i)8-s + (0.212 + 0.929i)9-s + (−0.422 − 0.529i)10-s + (0.143 − 0.629i)11-s + (0.0976 − 0.0470i)12-s + (−0.352 + 1.54i)13-s + (−0.631 − 0.318i)14-s + (0.0462 + 0.202i)15-s + (0.155 + 0.195i)16-s + (−1.07 + 0.515i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.430 + 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.430 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.430 + 0.902i$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :1/2),\ 0.430 + 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.911567 - 0.575291i\)
\(L(\frac12)\) \(\approx\) \(0.911567 - 0.575291i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.222 + 0.974i)T \)
7 \( 1 + (-0.635 + 2.56i)T \)
good3 \( 1 + (0.234 - 0.293i)T + (-0.667 - 2.92i)T^{2} \)
5 \( 1 + (-1.33 + 1.67i)T + (-1.11 - 4.87i)T^{2} \)
11 \( 1 + (-0.476 + 2.08i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (1.27 - 5.57i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + (4.41 - 2.12i)T + (10.5 - 13.2i)T^{2} \)
19 \( 1 + 2.06T + 19T^{2} \)
23 \( 1 + (-5.70 - 2.74i)T + (14.3 + 17.9i)T^{2} \)
29 \( 1 + (-7.15 + 3.44i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + 6.61T + 31T^{2} \)
37 \( 1 + (3.69 - 1.78i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 + (-3.90 + 4.89i)T + (-9.12 - 39.9i)T^{2} \)
43 \( 1 + (3.22 + 4.04i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (0.273 - 1.19i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (6.88 + 3.31i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + (1.71 + 2.15i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (-8.50 + 4.09i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 - 0.393T + 67T^{2} \)
71 \( 1 + (-11.3 - 5.46i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-0.959 - 4.20i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 + 12.1T + 79T^{2} \)
83 \( 1 + (1.51 + 6.63i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-0.216 - 0.948i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 - 3.19T + 97T^{2} \)
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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

−13.57602645803850703471638482786, −12.89022301034646216765615691533, −11.40016313002292032384205040618, −10.71944833541660838377952276141, −9.518018499993237464023560081984, −8.524379374147372309606015160948, −6.84815223393596566235179675915, −5.11741413795187214862309459236, −4.15844478956942588739506381173, −1.79656600168755395913370986933, 2.81702319216442834403032178908, 4.95241549357799817758247443284, 6.21993908544730044335389569950, 7.06417805266532080163797105263, 8.589382050320295589100770488492, 9.639685966007656305953778488678, 10.87462384581033656101101515797, 12.35533220910494495640833685606, 12.97967136978273020626634645712, 14.49759826610323280700361970821

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