Properties

Label 2-98-1.1-c9-0-27
Degree $2$
Conductor $98$
Sign $-1$
Analytic cond. $50.4735$
Root an. cond. $7.10447$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 16·2-s + 191.·3-s + 256·4-s + 33.5·5-s − 3.06e3·6-s − 4.09e3·8-s + 1.70e4·9-s − 537.·10-s − 5.15e4·11-s + 4.90e4·12-s + 1.13e5·13-s + 6.44e3·15-s + 6.55e4·16-s − 4.61e5·17-s − 2.73e5·18-s − 8.36e5·19-s + 8.59e3·20-s + 8.24e5·22-s + 2.99e5·23-s − 7.85e5·24-s − 1.95e6·25-s − 1.81e6·26-s − 4.96e5·27-s + 2.58e6·29-s − 1.03e5·30-s − 6.84e6·31-s − 1.04e6·32-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.36·3-s + 0.5·4-s + 0.0240·5-s − 0.966·6-s − 0.353·8-s + 0.868·9-s − 0.0169·10-s − 1.06·11-s + 0.683·12-s + 1.10·13-s + 0.0328·15-s + 0.250·16-s − 1.34·17-s − 0.614·18-s − 1.47·19-s + 0.0120·20-s + 0.750·22-s + 0.223·23-s − 0.483·24-s − 0.999·25-s − 0.777·26-s − 0.179·27-s + 0.678·29-s − 0.0232·30-s − 1.33·31-s − 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(50.4735\)
Root analytic conductor: \(7.10447\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 98,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 16T \)
7 \( 1 \)
good3 \( 1 - 191.T + 1.96e4T^{2} \)
5 \( 1 - 33.5T + 1.95e6T^{2} \)
11 \( 1 + 5.15e4T + 2.35e9T^{2} \)
13 \( 1 - 1.13e5T + 1.06e10T^{2} \)
17 \( 1 + 4.61e5T + 1.18e11T^{2} \)
19 \( 1 + 8.36e5T + 3.22e11T^{2} \)
23 \( 1 - 2.99e5T + 1.80e12T^{2} \)
29 \( 1 - 2.58e6T + 1.45e13T^{2} \)
31 \( 1 + 6.84e6T + 2.64e13T^{2} \)
37 \( 1 - 1.39e7T + 1.29e14T^{2} \)
41 \( 1 - 2.22e7T + 3.27e14T^{2} \)
43 \( 1 + 3.24e7T + 5.02e14T^{2} \)
47 \( 1 + 3.18e6T + 1.11e15T^{2} \)
53 \( 1 - 8.82e7T + 3.29e15T^{2} \)
59 \( 1 + 1.39e7T + 8.66e15T^{2} \)
61 \( 1 + 1.13e8T + 1.16e16T^{2} \)
67 \( 1 + 3.05e8T + 2.72e16T^{2} \)
71 \( 1 + 2.86e8T + 4.58e16T^{2} \)
73 \( 1 - 4.78e7T + 5.88e16T^{2} \)
79 \( 1 + 1.07e8T + 1.19e17T^{2} \)
83 \( 1 - 5.67e7T + 1.86e17T^{2} \)
89 \( 1 + 8.48e8T + 3.50e17T^{2} \)
97 \( 1 - 8.99e8T + 7.60e17T^{2} \)
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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

−11.24277345225287078426271840603, −10.33931967349023109269067270201, −9.039251960085061881400687804564, −8.472792814930664635751590879625, −7.52005837981465011104503782629, −6.11654860773958739787834790117, −4.14284049513747639397573803476, −2.76103622798157761524441454656, −1.83825219816964197246404820960, 0, 1.83825219816964197246404820960, 2.76103622798157761524441454656, 4.14284049513747639397573803476, 6.11654860773958739787834790117, 7.52005837981465011104503782629, 8.472792814930664635751590879625, 9.039251960085061881400687804564, 10.33931967349023109269067270201, 11.24277345225287078426271840603

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