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Properties

Label 2-30e2-36.23-c1-0-97
Degree $2$
Conductor $900$
Sign $-0.975 - 0.220i$
Analytic cond. $7.18653$
Root an. cond. $2.68077$
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  + (0.645 − 1.25i)2-s + (−1.67 − 0.424i)3-s + (−1.16 − 1.62i)4-s + (−1.61 + 1.83i)6-s + (2.78 − 1.60i)7-s + (−2.79 + 0.419i)8-s + (2.63 + 1.42i)9-s + (−1.56 − 2.70i)11-s + (1.26 + 3.22i)12-s + (0.923 − 1.59i)13-s + (−0.225 − 4.54i)14-s + (−1.27 + 3.79i)16-s − 5.85i·17-s + (3.49 − 2.39i)18-s + 2.24i·19-s + ⋯
L(s)  = 1  + (0.456 − 0.889i)2-s + (−0.969 − 0.245i)3-s + (−0.583 − 0.812i)4-s + (−0.660 + 0.750i)6-s + (1.05 − 0.608i)7-s + (−0.988 + 0.148i)8-s + (0.879 + 0.475i)9-s + (−0.471 − 0.816i)11-s + (0.366 + 0.930i)12-s + (0.256 − 0.443i)13-s + (−0.0602 − 1.21i)14-s + (−0.319 + 0.947i)16-s − 1.41i·17-s + (0.824 − 0.565i)18-s + 0.514i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.975 - 0.220i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (851, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ -0.975 - 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.123857 + 1.11180i\)
\(L(\frac12)\) \(\approx\) \(0.123857 + 1.11180i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.645 + 1.25i)T \)
3 \( 1 + (1.67 + 0.424i)T \)
5 \( 1 \)
good7 \( 1 + (-2.78 + 1.60i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.56 + 2.70i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.923 + 1.59i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.85iT - 17T^{2} \)
19 \( 1 - 2.24iT - 19T^{2} \)
23 \( 1 + (-1.87 + 3.24i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (6.90 - 3.98i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.03 + 0.599i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.66T + 37T^{2} \)
41 \( 1 + (0.208 + 0.120i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.61 - 2.66i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.28 - 3.96i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 13.0iT - 53T^{2} \)
59 \( 1 + (3.47 - 6.01i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.66 - 2.88i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.48 - 4.31i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 + 13.1T + 73T^{2} \)
79 \( 1 + (-11.6 + 6.72i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.623 + 1.07i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 14.3iT - 89T^{2} \)
97 \( 1 + (3.32 + 5.75i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.14314942686695003133297839841, −8.969594954542732765063625102810, −7.935330296395580068672886355731, −7.01432978876120749078513478428, −5.79790397959412287889377523875, −5.16877435767956935114860066893, −4.40454199564686573175313687789, −3.16554380864491186265076509142, −1.68835836982091010446409234279, −0.53424945488914418975632326989, 1.88356676653734061408778253781, 3.76252751590092011654825977291, 4.65379663332626406174026834222, 5.35409630762248763749389012676, 6.06329570703845252802120447108, 7.06247569790660461541135567468, 7.82203079679674763194450515863, 8.768019829709795936741554981143, 9.602514837262979590393256406402, 10.70534846147438821603953681786

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