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Properties

Label 2-30e2-36.23-c1-0-41
Degree $2$
Conductor $900$
Sign $0.754 - 0.656i$
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  + (−1.34 − 0.442i)2-s + (1.72 − 0.112i)3-s + (1.60 + 1.18i)4-s + (−2.37 − 0.613i)6-s + (−0.993 + 0.573i)7-s + (−1.63 − 2.30i)8-s + (2.97 − 0.388i)9-s + (0.629 + 1.09i)11-s + (2.91 + 1.87i)12-s + (−2.53 + 4.39i)13-s + (1.58 − 0.331i)14-s + (1.17 + 3.82i)16-s + 3.29i·17-s + (−4.16 − 0.793i)18-s + 3.62i·19-s + ⋯
L(s)  = 1  + (−0.949 − 0.312i)2-s + (0.997 − 0.0649i)3-s + (0.804 + 0.594i)4-s + (−0.968 − 0.250i)6-s + (−0.375 + 0.216i)7-s + (−0.578 − 0.815i)8-s + (0.991 − 0.129i)9-s + (0.189 + 0.328i)11-s + (0.841 + 0.540i)12-s + (−0.704 + 1.21i)13-s + (0.424 − 0.0885i)14-s + (0.294 + 0.955i)16-s + 0.799i·17-s + (−0.982 − 0.186i)18-s + 0.831i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 - 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.754 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.754 - 0.656i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (851, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ 0.754 - 0.656i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25396 + 0.468867i\)
\(L(\frac12)\) \(\approx\) \(1.25396 + 0.468867i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 + 0.442i)T \)
3 \( 1 + (-1.72 + 0.112i)T \)
5 \( 1 \)
good7 \( 1 + (0.993 - 0.573i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.629 - 1.09i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.53 - 4.39i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.29iT - 17T^{2} \)
19 \( 1 - 3.62iT - 19T^{2} \)
23 \( 1 + (-1.78 + 3.09i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.184 + 0.106i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-9.12 - 5.26i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 4.72T + 37T^{2} \)
41 \( 1 + (-5.81 - 3.35i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.45 - 1.41i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.534 - 0.925i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 9.12iT - 53T^{2} \)
59 \( 1 + (-4.87 + 8.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.24 + 9.08i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-12.8 - 7.42i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.68T + 71T^{2} \)
73 \( 1 + 9.08T + 73T^{2} \)
79 \( 1 + (-5.78 + 3.34i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.84 - 4.93i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 2.26iT - 89T^{2} \)
97 \( 1 + (-3.86 - 6.69i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.894538013191194779399004124890, −9.446185806686714475750476784215, −8.552088055957832678167531262334, −7.994377206422958061958334201218, −6.91052917950762894245151239160, −6.42617120377617727886873593469, −4.57574321424921909779166496763, −3.54325325106860475601108231334, −2.49378668153788920347585585077, −1.53772158822562359512665343614, 0.802417707567862931123364452774, 2.47269130783997994933973590914, 3.19655592589682047156198537395, 4.72752691585032442553355339067, 5.85660314155881131853046131112, 7.06101236681905022131319324586, 7.50652455512586515440266354424, 8.419438904260535598486873728009, 9.132184919165211998450170268846, 9.876881331336748646887971754724

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