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Properties

Label 2-98-7.4-c5-0-7
Degree $2$
Conductor $98$
Sign $-0.266 - 0.963i$
Analytic cond. $15.7176$
Root an. cond. $3.96454$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 + 3.46i)2-s + (5 − 8.66i)3-s + (−7.99 + 13.8i)4-s + (42 + 72.7i)5-s + 40·6-s − 63.9·8-s + (71.4 + 123. i)9-s + (−168 + 290. i)10-s + (168 − 290. i)11-s + (80 + 138. i)12-s − 584·13-s + 840.·15-s + (−128 − 221. i)16-s + (−729 + 1.26e3i)17-s + (−286 + 495. i)18-s + (235 + 407. i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.320 − 0.555i)3-s + (−0.249 + 0.433i)4-s + (0.751 + 1.30i)5-s + 0.453·6-s − 0.353·8-s + (0.294 + 0.509i)9-s + (−0.531 + 0.920i)10-s + (0.418 − 0.725i)11-s + (0.160 + 0.277i)12-s − 0.958·13-s + 0.963·15-s + (−0.125 − 0.216i)16-s + (−0.611 + 1.05i)17-s + (−0.208 + 0.360i)18-s + (0.149 + 0.258i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $-0.266 - 0.963i$
Analytic conductor: \(15.7176\)
Root analytic conductor: \(3.96454\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :5/2),\ -0.266 - 0.963i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.53257 + 2.01454i\)
\(L(\frac12)\) \(\approx\) \(1.53257 + 2.01454i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2 - 3.46i)T \)
7 \( 1 \)
good3 \( 1 + (-5 + 8.66i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (-42 - 72.7i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-168 + 290. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 584T + 3.71e5T^{2} \)
17 \( 1 + (729 - 1.26e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-235 - 407. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-2.10e3 - 3.63e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 4.86e3T + 2.05e7T^{2} \)
31 \( 1 + (3.68e3 - 6.38e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (7.16e3 + 1.24e4i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 6.22e3T + 1.15e8T^{2} \)
43 \( 1 - 3.70e3T + 1.47e8T^{2} \)
47 \( 1 + (906 + 1.56e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-1.86e4 + 3.22e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-1.71e4 + 2.97e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-1.22e4 - 2.11e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-8.72e3 + 1.51e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 2.82e4T + 1.80e9T^{2} \)
73 \( 1 + (-1.80e3 + 3.11e3i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (2.14e4 + 3.71e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 3.52e4T + 3.93e9T^{2} \)
89 \( 1 + (-1.33e4 - 2.31e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 1.69e4T + 8.58e9T^{2} \)
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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

−13.59211709189217075425446633789, −12.56550024395362911986304780062, −11.09129105595796595572184127104, −10.07722999777935460457955803517, −8.647096181778386197190966858699, −7.30263430629295054783955835797, −6.64656469033934855397053644493, −5.35684189062917643696109525949, −3.39861724708033735751536243205, −2.00664590801002533996666942040, 0.884845058604856041388485971121, 2.50529295948567086069025903300, 4.40404859765846826812380961120, 5.03261430322521054081143665524, 6.78169687999713946472967512441, 8.802608064368352772233621908433, 9.447447841024917438386079117027, 10.22749580723280521425237582562, 11.89767738394204554095450382699, 12.60107257665548804391872865371

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