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Properties

Label 2-30e2-9.4-c1-0-10
Degree $2$
Conductor $900$
Sign $0.389 + 0.921i$
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  + (−1.02 − 1.39i)3-s + (0.340 + 0.589i)7-s + (−0.880 + 2.86i)9-s + (0.840 + 1.45i)11-s + (2.57 − 4.45i)13-s + 1.31·17-s − 0.324·19-s + (0.470 − 1.08i)21-s + (1.89 − 3.28i)23-s + (4.90 − 1.72i)27-s + (4.32 + 7.48i)29-s + (2.07 − 3.58i)31-s + (1.16 − 2.66i)33-s − 1.35·37-s + (−8.85 + 1.00i)39-s + ⋯
L(s)  = 1  + (−0.594 − 0.804i)3-s + (0.128 + 0.222i)7-s + (−0.293 + 0.955i)9-s + (0.253 + 0.438i)11-s + (0.713 − 1.23i)13-s + 0.320·17-s − 0.0744·19-s + (0.102 − 0.235i)21-s + (0.395 − 0.684i)23-s + (0.943 − 0.331i)27-s + (0.802 + 1.39i)29-s + (0.372 − 0.644i)31-s + (0.202 − 0.464i)33-s − 0.222·37-s + (−1.41 + 0.160i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.389 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.389 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.389 + 0.921i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ 0.389 + 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07744 - 0.714147i\)
\(L(\frac12)\) \(\approx\) \(1.07744 - 0.714147i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.02 + 1.39i)T \)
5 \( 1 \)
good7 \( 1 + (-0.340 - 0.589i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.840 - 1.45i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.57 + 4.45i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.31T + 17T^{2} \)
19 \( 1 + 0.324T + 19T^{2} \)
23 \( 1 + (-1.89 + 3.28i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.32 - 7.48i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.07 + 3.58i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.35T + 37T^{2} \)
41 \( 1 + (-3.57 + 6.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.64 + 6.31i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.48 + 11.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.83T + 53T^{2} \)
59 \( 1 + (4.40 - 7.63i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.98 + 8.63i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.08 - 3.61i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.891T + 71T^{2} \)
73 \( 1 - 7.82T + 73T^{2} \)
79 \( 1 + (4.82 + 8.36i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.42 - 4.20i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 17.4T + 89T^{2} \)
97 \( 1 + (-4.46 - 7.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.34241925932581477066017683137, −8.879635611561638158993069728453, −8.246744896140675373566721543205, −7.30923893084694606359930032437, −6.54780122483716885318171364431, −5.61944691359690751215344287341, −4.90476207389005547419412042671, −3.44360227988192995022981670120, −2.16058408346592217329412982557, −0.808549741574295044698852557591, 1.21619461015966694845110778101, 3.08140494910483944347787991956, 4.12500489802437272922322327416, 4.81591126491943471313294374287, 6.05230574665014385400568576409, 6.51769086603915268885213792385, 7.79280606860444719320904667975, 8.811969278880578167624221960686, 9.483656134303564334406103768827, 10.26294465684669172536392296785

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