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Properties

Label 2-98-7.4-c3-0-2
Degree $2$
Conductor $98$
Sign $-0.991 + 0.126i$
Analytic cond. $5.78218$
Root an. cond. $2.40461$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)2-s + (−4 + 6.92i)3-s + (−1.99 + 3.46i)4-s + (7 + 12.1i)5-s − 15.9·6-s − 7.99·8-s + (−18.4 − 32.0i)9-s + (−14 + 24.2i)10-s + (14 − 24.2i)11-s + (−15.9 − 27.7i)12-s + 18·13-s − 112·15-s + (−8 − 13.8i)16-s + (−37 + 64.0i)17-s + (37 − 64.0i)18-s + (−40 − 69.2i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.769 + 1.33i)3-s + (−0.249 + 0.433i)4-s + (0.626 + 1.08i)5-s − 1.08·6-s − 0.353·8-s + (−0.685 − 1.18i)9-s + (−0.442 + 0.766i)10-s + (0.383 − 0.664i)11-s + (−0.384 − 0.666i)12-s + 0.384·13-s − 1.92·15-s + (−0.125 − 0.216i)16-s + (−0.527 + 0.914i)17-s + (0.484 − 0.839i)18-s + (−0.482 − 0.836i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(5.78218\)
Root analytic conductor: \(2.40461\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :3/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0880237 - 1.38707i\)
\(L(\frac12)\) \(\approx\) \(0.0880237 - 1.38707i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - 1.73i)T \)
7 \( 1 \)
good3 \( 1 + (4 - 6.92i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (-7 - 12.1i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-14 + 24.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 18T + 2.19e3T^{2} \)
17 \( 1 + (37 - 64.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (40 + 69.2i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-56 - 96.9i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 190T + 2.43e4T^{2} \)
31 \( 1 + (36 - 62.3i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-173 - 299. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 162T + 6.89e4T^{2} \)
43 \( 1 + 412T + 7.95e4T^{2} \)
47 \( 1 + (12 + 20.7i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (159 - 275. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-100 + 173. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-99 - 171. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-358 + 620. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 392T + 3.57e5T^{2} \)
73 \( 1 + (269 - 465. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (120 + 207. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.07e3T + 5.71e5T^{2} \)
89 \( 1 + (405 + 701. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.35e3T + 9.12e5T^{2} \)
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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

−14.23534341828736522045330952780, −13.20371472894862286672934778707, −11.49620874362813336611085072604, −10.79692042750257337933759432529, −9.847055857361340957179909248261, −8.621210715449709086793515304330, −6.66969774693845423182921770203, −5.95097404480670622678660658443, −4.62893088822233280156749678283, −3.26867368760505626465876652682, 0.818734667486362094071737159048, 2.04158525727052133325109303156, 4.63025257367326125683904459209, 5.81582514513634347794317059811, 6.88775802726927260919374450678, 8.483499966845826358509697583367, 9.703584351821489114195542910520, 11.12883665890889697133001382975, 12.14999325219835497599682072968, 12.76744530243510483156187460648

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