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Properties

Label 2-30e2-25.21-c1-0-10
Degree $2$
Conductor $900$
Sign $0.348 + 0.937i$
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  + (0.233 − 2.22i)5-s − 0.511·7-s + (0.564 − 1.73i)11-s + (1.89 + 5.82i)13-s + (2.09 − 1.51i)17-s + (3.93 − 2.85i)19-s + (2.04 − 6.30i)23-s + (−4.89 − 1.03i)25-s + (−6.46 − 4.70i)29-s + (3.95 − 2.87i)31-s + (−0.119 + 1.13i)35-s + (−2.29 − 7.07i)37-s + (−1.77 − 5.45i)41-s − 2.05·43-s + (6.43 + 4.67i)47-s + ⋯
L(s)  = 1  + (0.104 − 0.994i)5-s − 0.193·7-s + (0.170 − 0.523i)11-s + (0.524 + 1.61i)13-s + (0.506 − 0.368i)17-s + (0.902 − 0.655i)19-s + (0.427 − 1.31i)23-s + (−0.978 − 0.207i)25-s + (−1.20 − 0.872i)29-s + (0.709 − 0.515i)31-s + (−0.0201 + 0.192i)35-s + (−0.377 − 1.16i)37-s + (−0.276 − 0.851i)41-s − 0.312·43-s + (0.938 + 0.681i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.348 + 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.348 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.348 + 0.937i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ 0.348 + 0.937i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26331 - 0.878026i\)
\(L(\frac12)\) \(\approx\) \(1.26331 - 0.878026i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (-0.233 + 2.22i)T \)
good7 \( 1 + 0.511T + 7T^{2} \)
11 \( 1 + (-0.564 + 1.73i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-1.89 - 5.82i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.09 + 1.51i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-3.93 + 2.85i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-2.04 + 6.30i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (6.46 + 4.70i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-3.95 + 2.87i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.29 + 7.07i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.77 + 5.45i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 2.05T + 43T^{2} \)
47 \( 1 + (-6.43 - 4.67i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.07 - 0.781i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.11 - 9.59i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.47 + 4.53i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-11.5 + 8.42i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (4.34 + 3.15i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.07 - 3.31i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-1.06 - 0.775i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.738 + 0.536i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (3.63 - 11.1i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-5.98 - 4.34i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.628558132356628923356845958906, −9.173403491491453944997009912567, −8.461936153963919659633915460720, −7.41561046707650368367629868580, −6.45122918200374263890546778419, −5.55813919571520739857075859636, −4.57374983274287101808666081486, −3.71437499396646172914033654764, −2.21126873433061549955008787238, −0.805007162629954769946521962890, 1.49028878380302162565070412541, 3.12183880141708594069818604523, 3.54936756495585997116169913154, 5.22571923480006067712567252228, 5.88788118138998623570149698350, 6.95445770172368327775055589002, 7.64672917520746399899802707878, 8.471585071453193491318159470847, 9.852508234941619900672722646702, 10.05059869006779720473948014890

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