Properties

Label 2-98-49.22-c1-0-3
Degree $2$
Conductor $98$
Sign $0.707 + 0.706i$
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  + (−0.900 + 0.433i)2-s + (0.209 − 0.918i)3-s + (0.623 − 0.781i)4-s + (0.482 − 2.11i)5-s + (0.209 + 0.918i)6-s + (−2.20 − 1.45i)7-s + (−0.222 + 0.974i)8-s + (1.90 + 0.916i)9-s + (0.482 + 2.11i)10-s + (5.45 − 2.62i)11-s + (−0.587 − 0.736i)12-s + (−3.44 + 1.66i)13-s + (2.62 + 0.357i)14-s + (−1.84 − 0.886i)15-s + (−0.222 − 0.974i)16-s + (3.15 + 3.95i)17-s + ⋯
L(s)  = 1  + (−0.637 + 0.306i)2-s + (0.121 − 0.530i)3-s + (0.311 − 0.390i)4-s + (0.215 − 0.944i)5-s + (0.0856 + 0.375i)6-s + (−0.833 − 0.551i)7-s + (−0.0786 + 0.344i)8-s + (0.634 + 0.305i)9-s + (0.152 + 0.668i)10-s + (1.64 − 0.791i)11-s + (−0.169 − 0.212i)12-s + (−0.956 + 0.460i)13-s + (0.700 + 0.0956i)14-s + (−0.475 − 0.228i)15-s + (−0.0556 − 0.243i)16-s + (0.764 + 0.958i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.706i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.707 + 0.706i$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :1/2),\ 0.707 + 0.706i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.742527 - 0.307115i\)
\(L(\frac12)\) \(\approx\) \(0.742527 - 0.307115i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.900 - 0.433i)T \)
7 \( 1 + (2.20 + 1.45i)T \)
good3 \( 1 + (-0.209 + 0.918i)T + (-2.70 - 1.30i)T^{2} \)
5 \( 1 + (-0.482 + 2.11i)T + (-4.50 - 2.16i)T^{2} \)
11 \( 1 + (-5.45 + 2.62i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (3.44 - 1.66i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + (-3.15 - 3.95i)T + (-3.78 + 16.5i)T^{2} \)
19 \( 1 + 5.58T + 19T^{2} \)
23 \( 1 + (4.48 - 5.62i)T + (-5.11 - 22.4i)T^{2} \)
29 \( 1 + (0.0477 + 0.0599i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 - 0.373T + 31T^{2} \)
37 \( 1 + (-3.74 - 4.69i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + (0.332 - 1.45i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (0.193 + 0.847i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (-4.68 + 2.25i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-2.99 + 3.76i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (1.30 + 5.72i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (-4.78 - 6.00i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + 2.86T + 67T^{2} \)
71 \( 1 + (2.88 - 3.61i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (12.0 + 5.80i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 + 7.06T + 79T^{2} \)
83 \( 1 + (-8.70 - 4.19i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (10.7 + 5.18i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + 0.452T + 97T^{2} \)
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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

−13.72350464972584968710824293953, −12.77389968457939115287723495550, −11.83639352657401496426993262492, −10.23039692912500382397123474277, −9.367683836725538322019031148483, −8.307772699241552088831149084301, −7.04390176129274453179346481401, −6.05480190296412265303236017200, −4.12101843080151988256246250300, −1.43424991273567971812973133769, 2.60522072788790551476555136573, 4.12925506136444393910588379834, 6.37412090832295615074349208495, 7.20725018288177894944004976737, 9.028423127617663715841810177823, 9.814226289057691784086150641510, 10.44136468132412238428700834428, 12.00504276729171260965041986490, 12.59862214423551252593048735118, 14.46796796037444951824231496022

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