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Properties

Label 2-30e2-45.4-c1-0-10
Degree $2$
Conductor $900$
Sign $0.637 + 0.770i$
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  + (−1.68 − 0.403i)3-s + (3.32 − 1.91i)7-s + (2.67 + 1.35i)9-s + (0.853 + 1.47i)11-s + (−3.21 − 1.85i)13-s − 1.70i·17-s − 0.292·19-s + (−6.36 + 1.89i)21-s + (5.05 + 2.91i)23-s + (−3.95 − 3.36i)27-s + (4.33 + 7.50i)29-s + (−0.146 + 0.253i)31-s + (−0.841 − 2.83i)33-s − 11.9i·37-s + (4.66 + 4.41i)39-s + ⋯
L(s)  = 1  + (−0.972 − 0.232i)3-s + (1.25 − 0.724i)7-s + (0.891 + 0.452i)9-s + (0.257 + 0.445i)11-s + (−0.890 − 0.514i)13-s − 0.414i·17-s − 0.0671·19-s + (−1.38 + 0.412i)21-s + (1.05 + 0.608i)23-s + (−0.761 − 0.648i)27-s + (0.804 + 1.39i)29-s + (−0.0262 + 0.0455i)31-s + (−0.146 − 0.493i)33-s − 1.96i·37-s + (0.746 + 0.707i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.637 + 0.770i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ 0.637 + 0.770i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15701 - 0.544070i\)
\(L(\frac12)\) \(\approx\) \(1.15701 - 0.544070i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.68 + 0.403i)T \)
5 \( 1 \)
good7 \( 1 + (-3.32 + 1.91i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.853 - 1.47i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.21 + 1.85i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.70iT - 17T^{2} \)
19 \( 1 + 0.292T + 19T^{2} \)
23 \( 1 + (-5.05 - 2.91i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.33 - 7.50i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.146 - 0.253i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 11.9iT - 37T^{2} \)
41 \( 1 + (-3.48 + 6.02i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.21 + 1.85i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.62 - 0.936i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 11.6iT - 53T^{2} \)
59 \( 1 + (-5.83 + 10.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.48 + 12.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.26 - 4.77i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 15.9T + 71T^{2} \)
73 \( 1 - 8iT - 73T^{2} \)
79 \( 1 + (-1 - 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.2 - 5.91i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + (-3.17 + 1.83i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.19915027993394553519031722423, −9.291524903114887147487811261088, −8.054067615646644599699190470265, −7.29819812492190479095771652215, −6.79095039541998433193317259942, −5.23029213625042149042193059865, −5.03737787405304801707879887764, −3.85786918184931801964505152646, −2.08893395210047790583326522374, −0.838627071962337421370522109146, 1.23139192631601484777725761427, 2.63481096691837577504674821228, 4.37055699697297727032237175507, 4.84423004935869539613555180114, 5.85019888914253894251194725471, 6.61611481326859917986261664982, 7.70738995449633268462715235646, 8.570485137624285956736167433702, 9.455085985319386084332229034793, 10.36166826832301988910868336391

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