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Properties

Label 2-98-7.4-c3-0-5
Degree $2$
Conductor $98$
Sign $0.605 + 0.795i$
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 + (−2.5 + 4.33i)3-s + (−1.99 + 3.46i)4-s + (−4.5 − 7.79i)5-s + 10·6-s + 7.99·8-s + (0.999 + 1.73i)9-s + (−9 + 15.5i)10-s + (28.5 − 49.3i)11-s + (−10 − 17.3i)12-s + 70·13-s + 45.0·15-s + (−8 − 13.8i)16-s + (25.5 − 44.1i)17-s + (1.99 − 3.46i)18-s + (2.5 + 4.33i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.481 + 0.833i)3-s + (−0.249 + 0.433i)4-s + (−0.402 − 0.697i)5-s + 0.680·6-s + 0.353·8-s + (0.0370 + 0.0641i)9-s + (−0.284 + 0.492i)10-s + (0.781 − 1.35i)11-s + (−0.240 − 0.416i)12-s + 1.49·13-s + 0.774·15-s + (−0.125 − 0.216i)16-s + (0.363 − 0.630i)17-s + (0.0261 − 0.0453i)18-s + (0.0301 + 0.0522i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.605 + 0.795i$
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: \((2,\ 98,\ (\ :3/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.943948 - 0.467940i\)
\(L(\frac12)\) \(\approx\) \(0.943948 - 0.467940i\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.5 - 4.33i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (4.5 + 7.79i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-28.5 + 49.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 70T + 2.19e3T^{2} \)
17 \( 1 + (-25.5 + 44.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (34.5 + 59.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 114T + 2.43e4T^{2} \)
31 \( 1 + (-11.5 + 19.9i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-126.5 - 219. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 42T + 6.89e4T^{2} \)
43 \( 1 + 124T + 7.95e4T^{2} \)
47 \( 1 + (-100.5 - 174. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-196.5 + 340. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-109.5 + 189. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (354.5 + 614. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (209.5 - 362. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 96T + 3.57e5T^{2} \)
73 \( 1 + (156.5 - 271. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (230.5 + 399. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 588T + 5.71e5T^{2} \)
89 \( 1 + (508.5 + 880. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.83e3T + 9.12e5T^{2} \)
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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

−13.16259342003854485871889582038, −11.81024008987487486658733642658, −11.18782076071581031500476251414, −10.19176681065728094249705888841, −8.946764877558296449285425864421, −8.181234144018956430811129151642, −6.13690516830978083856269375870, −4.65532943160658943114519691213, −3.52422175297173536092827535390, −0.887277841584653983046864472412, 1.37839869608197073295916704853, 3.95753299742786935718457781720, 5.96103970597982696455029135360, 6.83363429475851773424809129261, 7.65084223796769923537305049754, 9.068772783441522075679089040429, 10.38126970777258602197910541954, 11.53567716790889727014184431275, 12.49072710563623397200206454905, 13.62372981054648011903105589250

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