Properties

Label 2-30e2-20.3-c1-0-33
Degree $2$
Conductor $900$
Sign $0.899 + 0.437i$
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  + (1.35 − 0.394i)2-s + (1.68 − 1.07i)4-s + (2.47 + 2.47i)7-s + (1.87 − 2.11i)8-s − 3.02i·11-s + (−0.363 − 0.363i)13-s + (4.34 + 2.38i)14-s + (1.70 − 3.61i)16-s + (−2.36 + 2.36i)17-s + 4.95·19-s + (−1.19 − 4.11i)22-s + (0.900 − 0.900i)23-s + (−0.636 − 0.350i)26-s + (6.83 + 1.53i)28-s + 3.50i·29-s + ⋯
L(s)  = 1  + (0.960 − 0.278i)2-s + (0.844 − 0.535i)4-s + (0.936 + 0.936i)7-s + (0.661 − 0.749i)8-s − 0.913i·11-s + (−0.100 − 0.100i)13-s + (1.16 + 0.638i)14-s + (0.426 − 0.904i)16-s + (−0.573 + 0.573i)17-s + 1.13·19-s + (−0.254 − 0.876i)22-s + (0.187 − 0.187i)23-s + (−0.124 − 0.0686i)26-s + (1.29 + 0.289i)28-s + 0.650i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.899 + 0.437i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ 0.899 + 0.437i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.11663 - 0.717124i\)
\(L(\frac12)\) \(\approx\) \(3.11663 - 0.717124i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.35 + 0.394i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-2.47 - 2.47i)T + 7iT^{2} \)
11 \( 1 + 3.02iT - 11T^{2} \)
13 \( 1 + (0.363 + 0.363i)T + 13iT^{2} \)
17 \( 1 + (2.36 - 2.36i)T - 17iT^{2} \)
19 \( 1 - 4.95T + 19T^{2} \)
23 \( 1 + (-0.900 + 0.900i)T - 23iT^{2} \)
29 \( 1 - 3.50iT - 29T^{2} \)
31 \( 1 - 3.85iT - 31T^{2} \)
37 \( 1 + (-0.363 + 0.363i)T - 37iT^{2} \)
41 \( 1 + 2.72T + 41T^{2} \)
43 \( 1 + (-3.92 + 3.92i)T - 43iT^{2} \)
47 \( 1 + (5.85 + 5.85i)T + 47iT^{2} \)
53 \( 1 + (-3.14 - 3.14i)T + 53iT^{2} \)
59 \( 1 - 8.68T + 59T^{2} \)
61 \( 1 + 15.2T + 61T^{2} \)
67 \( 1 + (3.92 + 3.92i)T + 67iT^{2} \)
71 \( 1 - 4.25iT - 71T^{2} \)
73 \( 1 + (9.28 + 9.28i)T + 73iT^{2} \)
79 \( 1 + 0.399T + 79T^{2} \)
83 \( 1 + (-0.199 + 0.199i)T - 83iT^{2} \)
89 \( 1 - 4.28iT - 89T^{2} \)
97 \( 1 + (6.73 - 6.73i)T - 97iT^{2} \)
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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

−10.39758754413174020381679771164, −9.155012712870011194476620619439, −8.398940696163749302314155236937, −7.39648538905054836118528681832, −6.33127035826830549459345962541, −5.46860101283383403998268494572, −4.89022706501003222190193363761, −3.63741152008250757304379843702, −2.64131653589828614064589527017, −1.45654348188610142030658698094, 1.55995192697763594548590717168, 2.84950676433086728954835731929, 4.19692777049697435928735112798, 4.68222628979461398594936300427, 5.65212992701810625212081354470, 6.86392408819761995806467810081, 7.45017917445660486879272218926, 8.091596902655034525059477231795, 9.417331306622624507415049689684, 10.32275794910023961154181447134

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