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Properties

Label 2-1098-9.4-c1-0-55
Degree $2$
Conductor $1098$
Sign $-0.250 - 0.968i$
Analytic cond. $8.76757$
Root an. cond. $2.96100$
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  + (−0.5 − 0.866i)2-s + (−1.53 − 0.806i)3-s + (−0.499 + 0.866i)4-s + (1.59 − 2.76i)5-s + (0.0679 + 1.73i)6-s + (−2.01 − 3.48i)7-s + 0.999·8-s + (1.69 + 2.47i)9-s − 3.19·10-s + (−2.90 − 5.03i)11-s + (1.46 − 0.924i)12-s + (−2.04 + 3.54i)13-s + (−2.01 + 3.48i)14-s + (−4.67 + 2.95i)15-s + (−0.5 − 0.866i)16-s − 3.84·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.884 − 0.465i)3-s + (−0.249 + 0.433i)4-s + (0.713 − 1.23i)5-s + (0.0277 + 0.706i)6-s + (−0.761 − 1.31i)7-s + 0.353·8-s + (0.566 + 0.824i)9-s − 1.00·10-s + (−0.877 − 1.51i)11-s + (0.422 − 0.266i)12-s + (−0.566 + 0.982i)13-s + (−0.538 + 0.932i)14-s + (−1.20 + 0.761i)15-s + (−0.125 − 0.216i)16-s − 0.932·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.250 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.250 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1098\)    =    \(2 \cdot 3^{2} \cdot 61\)
Sign: $-0.250 - 0.968i$
Analytic conductor: \(8.76757\)
Root analytic conductor: \(2.96100\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1098} (733, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1098,\ (\ :1/2),\ -0.250 - 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5118972814\)
\(L(\frac12)\) \(\approx\) \(0.5118972814\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (1.53 + 0.806i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-1.59 + 2.76i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.01 + 3.48i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.90 + 5.03i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.04 - 3.54i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.84T + 17T^{2} \)
19 \( 1 - 3.74T + 19T^{2} \)
23 \( 1 + (-3.09 + 5.35i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.82 + 3.16i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.97 - 5.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.36T + 37T^{2} \)
41 \( 1 + (-2.02 + 3.51i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.31 + 9.20i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.78 - 8.29i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6.91T + 53T^{2} \)
59 \( 1 + (4.60 - 7.97i)T + (-29.5 - 51.0i)T^{2} \)
67 \( 1 + (0.494 - 0.857i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.71T + 71T^{2} \)
73 \( 1 + 15.9T + 73T^{2} \)
79 \( 1 + (-6.03 - 10.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.758 + 1.31i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 0.883T + 89T^{2} \)
97 \( 1 + (-2.85 - 4.94i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.324548671050771141065698158836, −8.680817695723821590553931174446, −7.57383530520536516156876466759, −6.78554625247136334939360323690, −5.80453278317862720873278572280, −4.91638410824573627313768210879, −4.06323515635081365381489479997, −2.54656313943366307167179383508, −1.13477466060181891643063108610, −0.31952976631360448718259655911, 2.20440622425686146578572097486, 3.15023124305238797847904287884, 4.85167265592084476467171959266, 5.54082600505629664610158316971, 6.16199181465808209992094924342, 7.02703425395360669618675039297, 7.62500751501548542659568987978, 9.206805348150377855300190466991, 9.708200233259377748846739452361, 10.14573521455820717719867220843

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