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Properties

Label 2-98-49.43-c1-0-4
Degree $2$
Conductor $98$
Sign $0.943 + 0.332i$
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  + (0.623 − 0.781i)2-s + (2.57 + 1.23i)3-s + (−0.222 − 0.974i)4-s + (−2.64 − 1.27i)5-s + (2.57 − 1.23i)6-s + (−2.49 + 0.891i)7-s + (−0.900 − 0.433i)8-s + (3.21 + 4.03i)9-s + (−2.64 + 1.27i)10-s + (−2.98 + 3.73i)11-s + (0.635 − 2.78i)12-s + (3.28 − 4.11i)13-s + (−0.856 + 2.50i)14-s + (−5.22 − 6.54i)15-s + (−0.900 + 0.433i)16-s + (0.472 − 2.07i)17-s + ⋯
L(s)  = 1  + (0.440 − 0.552i)2-s + (1.48 + 0.715i)3-s + (−0.111 − 0.487i)4-s + (−1.18 − 0.569i)5-s + (1.05 − 0.505i)6-s + (−0.941 + 0.336i)7-s + (−0.318 − 0.153i)8-s + (1.07 + 1.34i)9-s + (−0.835 + 0.402i)10-s + (−0.898 + 1.12i)11-s + (0.183 − 0.803i)12-s + (0.909 − 1.14i)13-s + (−0.228 + 0.669i)14-s + (−1.34 − 1.69i)15-s + (−0.225 + 0.108i)16-s + (0.114 − 0.502i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.332i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 + 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.943 + 0.332i$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :1/2),\ 0.943 + 0.332i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43368 - 0.244978i\)
\(L(\frac12)\) \(\approx\) \(1.43368 - 0.244978i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.623 + 0.781i)T \)
7 \( 1 + (2.49 - 0.891i)T \)
good3 \( 1 + (-2.57 - 1.23i)T + (1.87 + 2.34i)T^{2} \)
5 \( 1 + (2.64 + 1.27i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (2.98 - 3.73i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (-3.28 + 4.11i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + (-0.472 + 2.07i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 - 3.82T + 19T^{2} \)
23 \( 1 + (-0.351 - 1.54i)T + (-20.7 + 9.97i)T^{2} \)
29 \( 1 + (0.132 - 0.582i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + 0.0848T + 31T^{2} \)
37 \( 1 + (1.14 - 5.00i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (6.48 + 3.12i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-3.87 + 1.86i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-0.628 + 0.788i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (0.641 + 2.80i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (1.82 - 0.878i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (0.826 - 3.62i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + 11.9T + 67T^{2} \)
71 \( 1 + (-0.746 - 3.27i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-4.76 - 5.97i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 - 10.8T + 79T^{2} \)
83 \( 1 + (-6.72 - 8.42i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (7.48 + 9.38i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 - 10.5T + 97T^{2} \)
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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

−13.70078828684281008743540108845, −12.94085519046566503757829829504, −12.00786745479277598956589824554, −10.43429536554866303709613866008, −9.578830826474184872473576209605, −8.508309635778437965772586570319, −7.51816097988212338425788808859, −5.12268201845879935875097512037, −3.79083971535151921980422814011, −2.90511231420836474517133802064, 3.09113984392995477379737988006, 3.79921478374821276234375927021, 6.35054048485403421826655856406, 7.38729549837990878186957289322, 8.149906509383811919267432569764, 9.150535294394742327270825545860, 10.92248877395674347458477566780, 12.24173519365485090045293170297, 13.42226264017860872903804307423, 13.81860165945813018511404444126

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