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Properties

Label 2-98-7.6-c4-0-0
Degree $2$
Conductor $98$
Sign $0.156 - 0.987i$
Analytic cond. $10.1302$
Root an. cond. $3.18280$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.82·2-s − 15.9i·3-s + 8.00·4-s + 27.8i·5-s + 45.2i·6-s − 22.6·8-s − 174.·9-s − 78.7i·10-s − 139.·11-s − 127. i·12-s + 101. i·13-s + 445.·15-s + 64.0·16-s + 542. i·17-s + 494.·18-s − 139. i·19-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.77i·3-s + 0.500·4-s + 1.11i·5-s + 1.25i·6-s − 0.353·8-s − 2.15·9-s − 0.787i·10-s − 1.15·11-s − 0.888i·12-s + 0.603i·13-s + 1.97·15-s + 0.250·16-s + 1.87i·17-s + 1.52·18-s − 0.387i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.156 - 0.987i$
Analytic conductor: \(10.1302\)
Root analytic conductor: \(3.18280\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :2),\ 0.156 - 0.987i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.337733 + 0.288415i\)
\(L(\frac12)\) \(\approx\) \(0.337733 + 0.288415i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2.82T \)
7 \( 1 \)
good3 \( 1 + 15.9iT - 81T^{2} \)
5 \( 1 - 27.8iT - 625T^{2} \)
11 \( 1 + 139.T + 1.46e4T^{2} \)
13 \( 1 - 101. iT - 2.85e4T^{2} \)
17 \( 1 - 542. iT - 8.35e4T^{2} \)
19 \( 1 + 139. iT - 1.30e5T^{2} \)
23 \( 1 - 229.T + 2.79e5T^{2} \)
29 \( 1 + 383.T + 7.07e5T^{2} \)
31 \( 1 - 397. iT - 9.23e5T^{2} \)
37 \( 1 + 898.T + 1.87e6T^{2} \)
41 \( 1 - 657. iT - 2.82e6T^{2} \)
43 \( 1 - 1.15e3T + 3.41e6T^{2} \)
47 \( 1 + 1.29e3iT - 4.87e6T^{2} \)
53 \( 1 + 5.16e3T + 7.89e6T^{2} \)
59 \( 1 - 4.15e3iT - 1.21e7T^{2} \)
61 \( 1 - 1.84e3iT - 1.38e7T^{2} \)
67 \( 1 + 6.31e3T + 2.01e7T^{2} \)
71 \( 1 - 1.26e3T + 2.54e7T^{2} \)
73 \( 1 + 3.99e3iT - 2.83e7T^{2} \)
79 \( 1 + 1.06e4T + 3.89e7T^{2} \)
83 \( 1 - 5.75e3iT - 4.74e7T^{2} \)
89 \( 1 - 4.18e3iT - 6.27e7T^{2} \)
97 \( 1 + 3.81e3iT - 8.85e7T^{2} \)
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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

−13.26393144563812473529900925060, −12.43606155399627021964971468392, −11.22223231339560746606456391639, −10.46074666685913892294238466326, −8.666098285810390041820364983721, −7.68059906483042590801752946271, −6.88342580229423453629458982456, −5.96654984231932395010086976932, −2.88266363079974735994877837361, −1.68887264766193480416881110644, 0.24786432842720886198987614463, 3.00676247711538200237229246945, 4.74174624086382799585636322581, 5.48041491906414501546398360681, 7.78847365964386137030026195798, 8.917481187059213872296197122585, 9.606054699816702805520854115253, 10.53253385726554078150090923807, 11.49045438362344790552224537707, 12.83616963315262229440993533262

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