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Properties

Label 2-30e2-45.34-c1-0-15
Degree $2$
Conductor $900$
Sign $0.00244 + 0.999i$
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  + (1.07 − 1.36i)3-s + (0.0748 + 0.0432i)7-s + (−0.704 − 2.91i)9-s + (0.456 − 0.791i)11-s + (2.27 − 1.31i)13-s − 2.08i·17-s − 4.93·19-s + (0.139 − 0.0555i)21-s + (7.34 − 4.23i)23-s + (−4.72 − 2.16i)27-s + (1.19 − 2.07i)29-s + (−1.81 − 3.13i)31-s + (−0.587 − 1.46i)33-s + 5.85i·37-s + (0.649 − 4.49i)39-s + ⋯
L(s)  = 1  + (0.618 − 0.785i)3-s + (0.0283 + 0.0163i)7-s + (−0.234 − 0.972i)9-s + (0.137 − 0.238i)11-s + (0.630 − 0.364i)13-s − 0.506i·17-s − 1.13·19-s + (0.0303 − 0.0121i)21-s + (1.53 − 0.883i)23-s + (−0.908 − 0.416i)27-s + (0.222 − 0.385i)29-s + (−0.325 − 0.563i)31-s + (−0.102 − 0.255i)33-s + 0.962i·37-s + (0.103 − 0.720i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00244 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00244 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.00244 + 0.999i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ 0.00244 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31667 - 1.31345i\)
\(L(\frac12)\) \(\approx\) \(1.31667 - 1.31345i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.07 + 1.36i)T \)
5 \( 1 \)
good7 \( 1 + (-0.0748 - 0.0432i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.456 + 0.791i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.27 + 1.31i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.08iT - 17T^{2} \)
19 \( 1 + 4.93T + 19T^{2} \)
23 \( 1 + (-7.34 + 4.23i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.19 + 2.07i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.81 + 3.13i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.85iT - 37T^{2} \)
41 \( 1 + (3.32 + 5.75i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-7.14 - 4.12i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.32 - 1.34i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 5.73iT - 53T^{2} \)
59 \( 1 + (-6.16 - 10.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.16 + 5.48i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.33 - 3.08i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 - 5.31iT - 73T^{2} \)
79 \( 1 + (6.72 - 11.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.26 - 3.03i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 8.13T + 89T^{2} \)
97 \( 1 + (-9.61 - 5.55i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.784832958559721540986487286919, −8.728338775717346674348541620160, −8.436170118008165886638609621437, −7.30882047771943549506880969488, −6.60965109899834251229347972939, −5.73387113180631621073775241413, −4.41566527012407761828466090624, −3.27315736089576791392955641615, −2.30374440003603519653904350212, −0.864828846757849394838894807214, 1.73867986088937188061383417030, 3.05938935433255015356529487394, 3.99356696757078226951677921024, 4.84153577092479601959559535361, 5.90268726118679087260557216846, 7.00225395598928391761885510646, 7.936195097324640292662889318481, 8.958861358937053355997318816712, 9.179157847757784102109304022268, 10.48250459369020868633771773611

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