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Properties

Label 2-98-49.36-c1-0-2
Degree $2$
Conductor $98$
Sign $0.175 - 0.984i$
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 + (1.37 + 1.72i)3-s + (−0.900 + 0.433i)4-s + (−0.294 − 0.369i)5-s + (−1.37 + 1.72i)6-s + (−0.972 − 2.46i)7-s + (−0.623 − 0.781i)8-s + (−0.412 + 1.80i)9-s + (0.294 − 0.369i)10-s + (−0.105 − 0.461i)11-s + (−1.98 − 0.955i)12-s + (0.0313 + 0.137i)13-s + (2.18 − 1.49i)14-s + (0.231 − 1.01i)15-s + (0.623 − 0.781i)16-s + (6.09 + 2.93i)17-s + ⋯
L(s)  = 1  + (0.157 + 0.689i)2-s + (0.792 + 0.994i)3-s + (−0.450 + 0.216i)4-s + (−0.131 − 0.165i)5-s + (−0.560 + 0.703i)6-s + (−0.367 − 0.929i)7-s + (−0.220 − 0.276i)8-s + (−0.137 + 0.602i)9-s + (0.0932 − 0.116i)10-s + (−0.0317 − 0.139i)11-s + (−0.572 − 0.275i)12-s + (0.00869 + 0.0380i)13-s + (0.583 − 0.399i)14-s + (0.0598 − 0.262i)15-s + (0.155 − 0.195i)16-s + (1.47 + 0.711i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.175 - 0.984i$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :1/2),\ 0.175 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.934359 + 0.782153i\)
\(L(\frac12)\) \(\approx\) \(0.934359 + 0.782153i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.972 + 2.46i)T \)
good3 \( 1 + (-1.37 - 1.72i)T + (-0.667 + 2.92i)T^{2} \)
5 \( 1 + (0.294 + 0.369i)T + (-1.11 + 4.87i)T^{2} \)
11 \( 1 + (0.105 + 0.461i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (-0.0313 - 0.137i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + (-6.09 - 2.93i)T + (10.5 + 13.2i)T^{2} \)
19 \( 1 + 4.64T + 19T^{2} \)
23 \( 1 + (3.73 - 1.79i)T + (14.3 - 17.9i)T^{2} \)
29 \( 1 + (4.42 + 2.12i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 - 0.381T + 31T^{2} \)
37 \( 1 + (9.73 + 4.68i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (-6.40 - 8.03i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (4.50 - 5.64i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (-0.623 - 2.73i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-11.3 + 5.44i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + (7.83 - 9.83i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (-3.49 - 1.68i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 + 8.63T + 67T^{2} \)
71 \( 1 + (2.99 - 1.44i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-1.41 + 6.19i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 - 9.98T + 79T^{2} \)
83 \( 1 + (-2.43 + 10.6i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-0.676 + 2.96i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + 3.36T + 97T^{2} \)
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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

−14.43260432640949790549049424676, −13.45872898666953022906301023486, −12.30489794140234708173830080281, −10.52097124740080675066425507504, −9.793674563937412800596168130954, −8.601531882493345924051601939234, −7.63193887832852064229082620973, −6.09421370422275820207113235167, −4.39380538502797712636194103151, −3.50845291057178766919656317037, 2.06781795112045790270722148753, 3.35017140725680172569522535881, 5.46920846853225363353823427997, 7.03761841582629381276304504509, 8.267819513783762700965256303906, 9.234352785605502717926741251106, 10.51736761264115408564108004986, 12.02740911273869323864375590991, 12.52508761579754992646672198514, 13.56464231774992285738322600824

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