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Properties

Label 2-98-49.46-c1-0-3
Degree $2$
Conductor $98$
Sign $-0.145 + 0.989i$
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  + (0.365 − 0.930i)2-s + (0.0722 − 0.964i)3-s + (−0.733 − 0.680i)4-s + (−3.07 − 2.09i)5-s + (−0.871 − 0.419i)6-s + (2.50 + 0.842i)7-s + (−0.900 + 0.433i)8-s + (2.04 + 0.307i)9-s + (−3.07 + 2.09i)10-s + (0.852 − 0.128i)11-s + (−0.709 + 0.657i)12-s + (2.83 + 3.54i)13-s + (1.70 − 2.02i)14-s + (−2.24 + 2.81i)15-s + (0.0747 + 0.997i)16-s + (−0.441 − 0.136i)17-s + ⋯
L(s)  = 1  + (0.258 − 0.658i)2-s + (0.0417 − 0.556i)3-s + (−0.366 − 0.340i)4-s + (−1.37 − 0.936i)5-s + (−0.355 − 0.171i)6-s + (0.947 + 0.318i)7-s + (−0.318 + 0.153i)8-s + (0.680 + 0.102i)9-s + (−0.970 + 0.661i)10-s + (0.257 − 0.0387i)11-s + (−0.204 + 0.189i)12-s + (0.784 + 0.984i)13-s + (0.454 − 0.541i)14-s + (−0.578 + 0.725i)15-s + (0.0186 + 0.249i)16-s + (−0.107 − 0.0330i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.145 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.145 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $-0.145 + 0.989i$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :1/2),\ -0.145 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.672750 - 0.778908i\)
\(L(\frac12)\) \(\approx\) \(0.672750 - 0.778908i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.365 + 0.930i)T \)
7 \( 1 + (-2.50 - 0.842i)T \)
good3 \( 1 + (-0.0722 + 0.964i)T + (-2.96 - 0.447i)T^{2} \)
5 \( 1 + (3.07 + 2.09i)T + (1.82 + 4.65i)T^{2} \)
11 \( 1 + (-0.852 + 0.128i)T + (10.5 - 3.24i)T^{2} \)
13 \( 1 + (-2.83 - 3.54i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + (0.441 + 0.136i)T + (14.0 + 9.57i)T^{2} \)
19 \( 1 + (2.39 + 4.15i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.97 - 0.917i)T + (19.0 - 12.9i)T^{2} \)
29 \( 1 + (-1.91 - 8.39i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 + (3.17 - 5.50i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.89 + 5.47i)T + (2.76 - 36.8i)T^{2} \)
41 \( 1 + (2.40 - 1.15i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (9.27 + 4.46i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-0.0212 + 0.0541i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (3.50 + 3.24i)T + (3.96 + 52.8i)T^{2} \)
59 \( 1 + (-8.57 + 5.84i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (10.6 - 9.87i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (-0.386 + 0.669i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.541 - 2.37i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-1.02 - 2.61i)T + (-53.5 + 49.6i)T^{2} \)
79 \( 1 + (-1.59 - 2.76i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.787 - 0.987i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (-1.62 - 0.245i)T + (85.0 + 26.2i)T^{2} \)
97 \( 1 - 6.58T + 97T^{2} \)
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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

−13.34925061041495257163930693831, −12.41785103551987012771348718202, −11.71570782824939116632920062036, −10.89492311170869942384937378153, −9.006651790931865629087233713994, −8.279843605850909745065825828382, −6.96749299730595983105365405847, −4.91804956226904378900981961861, −3.99422371271866491127937634797, −1.51981947677742087973078348068, 3.65556671487113610466631152607, 4.45168793181720368488593412255, 6.30134355426150376880494159837, 7.69357578379669020075561944531, 8.198631936690390349753956809524, 10.07732215600653168488557538323, 11.03476439704262300169435199983, 11.99513517780587774504268277794, 13.37757340226506467919444337001, 14.71699401176742306045106374641

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