Properties

Label 2-98-49.32-c1-0-2
Degree $2$
Conductor $98$
Sign $0.794 + 0.607i$
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  + (−0.0747 − 0.997i)2-s + (1.52 + 0.471i)3-s + (−0.988 + 0.149i)4-s + (−0.144 + 0.133i)5-s + (0.355 − 1.55i)6-s + (2.44 − 1.00i)7-s + (0.222 + 0.974i)8-s + (−0.364 − 0.248i)9-s + (0.144 + 0.133i)10-s + (−3.03 + 2.07i)11-s + (−1.58 − 0.238i)12-s + (−1.91 − 0.921i)13-s + (−1.18 − 2.36i)14-s + (−0.283 + 0.136i)15-s + (0.955 − 0.294i)16-s + (1.42 + 3.63i)17-s + ⋯
L(s)  = 1  + (−0.0528 − 0.705i)2-s + (0.882 + 0.272i)3-s + (−0.494 + 0.0745i)4-s + (−0.0645 + 0.0598i)5-s + (0.145 − 0.636i)6-s + (0.925 − 0.379i)7-s + (0.0786 + 0.344i)8-s + (−0.121 − 0.0827i)9-s + (0.0456 + 0.0423i)10-s + (−0.915 + 0.624i)11-s + (−0.456 − 0.0688i)12-s + (−0.530 − 0.255i)13-s + (−0.316 − 0.632i)14-s + (−0.0732 + 0.0352i)15-s + (0.238 − 0.0736i)16-s + (0.346 + 0.882i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.794 + 0.607i$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :1/2),\ 0.794 + 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11289 - 0.376981i\)
\(L(\frac12)\) \(\approx\) \(1.11289 - 0.376981i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0747 + 0.997i)T \)
7 \( 1 + (-2.44 + 1.00i)T \)
good3 \( 1 + (-1.52 - 0.471i)T + (2.47 + 1.68i)T^{2} \)
5 \( 1 + (0.144 - 0.133i)T + (0.373 - 4.98i)T^{2} \)
11 \( 1 + (3.03 - 2.07i)T + (4.01 - 10.2i)T^{2} \)
13 \( 1 + (1.91 + 0.921i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (-1.42 - 3.63i)T + (-12.4 + 11.5i)T^{2} \)
19 \( 1 + (2.48 + 4.29i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.79 - 4.56i)T + (-16.8 - 15.6i)T^{2} \)
29 \( 1 + (0.499 - 0.625i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (0.00876 - 0.0151i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.77 - 1.32i)T + (35.3 + 10.9i)T^{2} \)
41 \( 1 + (0.431 + 1.88i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (-2.17 + 9.53i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (0.497 + 6.63i)T + (-46.4 + 7.00i)T^{2} \)
53 \( 1 + (-1.69 + 0.256i)T + (50.6 - 15.6i)T^{2} \)
59 \( 1 + (10.5 + 9.74i)T + (4.40 + 58.8i)T^{2} \)
61 \( 1 + (-9.87 - 1.48i)T + (58.2 + 17.9i)T^{2} \)
67 \( 1 + (3.63 - 6.29i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.28 - 7.88i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (0.0400 - 0.534i)T + (-72.1 - 10.8i)T^{2} \)
79 \( 1 + (-3.58 - 6.21i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.11 + 2.46i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (-11.0 - 7.55i)T + (32.5 + 82.8i)T^{2} \)
97 \( 1 + 14.8T + 97T^{2} \)
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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

−13.76629786757351256948879513566, −12.85640283888682946533121170058, −11.58568371256239199589915343508, −10.57103866025929649205023434608, −9.564679835587160551822393867829, −8.391420866762324497520707331449, −7.51007340057050973522263166545, −5.22700500501401340544805151004, −3.83713018786563063392669706203, −2.29026062402909445993259642603, 2.57528810417995109468605858266, 4.65819487938708336282582912873, 5.99140668536068573652854551172, 7.82750710136583091619167677143, 8.114101517020926541386122509758, 9.319400407338978333495331973993, 10.75326708470008984576735586985, 12.11240434158948459943096651397, 13.32344971995344446645363933403, 14.36619206407316825891850330712

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