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Properties

Label 2-30e2-36.11-c1-0-22
Degree $2$
Conductor $900$
Sign $0.830 + 0.557i$
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  + (0.0449 − 1.41i)2-s + (−0.589 − 1.62i)3-s + (−1.99 − 0.126i)4-s + (−2.32 + 0.759i)6-s + (2.50 + 1.44i)7-s + (−0.269 + 2.81i)8-s + (−2.30 + 1.91i)9-s + (0.395 − 0.684i)11-s + (0.968 + 3.32i)12-s + (2.33 + 4.04i)13-s + (2.15 − 3.47i)14-s + (3.96 + 0.506i)16-s + 5.89i·17-s + (2.60 + 3.34i)18-s + 4.55i·19-s + ⋯
L(s)  = 1  + (0.0317 − 0.999i)2-s + (−0.340 − 0.940i)3-s + (−0.997 − 0.0634i)4-s + (−0.950 + 0.310i)6-s + (0.945 + 0.546i)7-s + (−0.0951 + 0.995i)8-s + (−0.768 + 0.639i)9-s + (0.119 − 0.206i)11-s + (0.279 + 0.960i)12-s + (0.647 + 1.12i)13-s + (0.575 − 0.927i)14-s + (0.991 + 0.126i)16-s + 1.42i·17-s + (0.614 + 0.788i)18-s + 1.04i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 + 0.557i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.830 + 0.557i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ 0.830 + 0.557i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17804 - 0.358731i\)
\(L(\frac12)\) \(\approx\) \(1.17804 - 0.358731i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0449 + 1.41i)T \)
3 \( 1 + (0.589 + 1.62i)T \)
5 \( 1 \)
good7 \( 1 + (-2.50 - 1.44i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.395 + 0.684i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.33 - 4.04i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 5.89iT - 17T^{2} \)
19 \( 1 - 4.55iT - 19T^{2} \)
23 \( 1 + (-0.666 - 1.15i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.29 + 1.32i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.26 + 1.30i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.44T + 37T^{2} \)
41 \( 1 + (5.91 - 3.41i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-10.3 - 5.95i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.50 - 2.60i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 1.70iT - 53T^{2} \)
59 \( 1 + (5.29 + 9.17i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.869 - 1.50i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.52 + 3.76i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.88T + 71T^{2} \)
73 \( 1 + 6.50T + 73T^{2} \)
79 \( 1 + (-3.30 - 1.91i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.81 + 8.33i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 0.00741iT - 89T^{2} \)
97 \( 1 + (3.31 - 5.74i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.29954124151952005432377609343, −9.096989830311111769566031040218, −8.380203779613911869087616689213, −7.82699672277729887121853661905, −6.37032001244758195679792198628, −5.68091593401777009192165840501, −4.61614167006795153186560037961, −3.51438586058518114430210363437, −1.99213371516862108796276106531, −1.46760674306086974784504402086, 0.67446156898122512701642963689, 3.14607326397759766884264186175, 4.25010764503755713987251000605, 5.02217043687962280794706985968, 5.59573297683979782421267446106, 6.82581784601360009473095718206, 7.56466609937442808836043933380, 8.601771079291130283728332561991, 9.136852039672442989330230606304, 10.19120129051796717784115733195

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