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Properties

Label 2-30e2-20.7-c1-0-42
Degree $2$
Conductor $900$
Sign $-0.993 - 0.117i$
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  + (0.760 − 1.19i)2-s + (−0.844 − 1.81i)4-s + (−0.611 + 0.611i)7-s + (−2.80 − 0.371i)8-s − 5.12i·11-s + (−1.76 + 1.76i)13-s + (0.264 + 1.19i)14-s + (−2.57 + 3.06i)16-s + (−3.76 − 3.76i)17-s − 1.22·19-s + (−6.11 − 3.89i)22-s + (−1.07 − 1.07i)23-s + (0.761 + 3.43i)26-s + (1.62 + 0.592i)28-s + 0.864i·29-s + ⋯
L(s)  = 1  + (0.537 − 0.843i)2-s + (−0.422 − 0.906i)4-s + (−0.231 + 0.231i)7-s + (−0.991 − 0.131i)8-s − 1.54i·11-s + (−0.488 + 0.488i)13-s + (0.0706 + 0.319i)14-s + (−0.643 + 0.765i)16-s + (−0.912 − 0.912i)17-s − 0.280·19-s + (−1.30 − 0.831i)22-s + (−0.224 − 0.224i)23-s + (0.149 + 0.674i)26-s + (0.307 + 0.111i)28-s + 0.160i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.117i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.993 - 0.117i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ -0.993 - 0.117i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0658712 + 1.11692i\)
\(L(\frac12)\) \(\approx\) \(0.0658712 + 1.11692i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.760 + 1.19i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (0.611 - 0.611i)T - 7iT^{2} \)
11 \( 1 + 5.12iT - 11T^{2} \)
13 \( 1 + (1.76 - 1.76i)T - 13iT^{2} \)
17 \( 1 + (3.76 + 3.76i)T + 17iT^{2} \)
19 \( 1 + 1.22T + 19T^{2} \)
23 \( 1 + (1.07 + 1.07i)T + 23iT^{2} \)
29 \( 1 - 0.864iT - 29T^{2} \)
31 \( 1 + 7.81iT - 31T^{2} \)
37 \( 1 + (-1.76 - 1.76i)T + 37iT^{2} \)
41 \( 1 + 5.52T + 41T^{2} \)
43 \( 1 + (6.20 + 6.20i)T + 43iT^{2} \)
47 \( 1 + (-2.29 + 2.29i)T - 47iT^{2} \)
53 \( 1 + (2.62 - 2.62i)T - 53iT^{2} \)
59 \( 1 - 0.528T + 59T^{2} \)
61 \( 1 - 4.98T + 61T^{2} \)
67 \( 1 + (-6.20 + 6.20i)T - 67iT^{2} \)
71 \( 1 - 8.10iT - 71T^{2} \)
73 \( 1 + (-2.25 + 2.25i)T - 73iT^{2} \)
79 \( 1 - 15.9T + 79T^{2} \)
83 \( 1 + (7.95 + 7.95i)T + 83iT^{2} \)
89 \( 1 - 7.25iT - 89T^{2} \)
97 \( 1 + (0.793 + 0.793i)T + 97iT^{2} \)
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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

−9.726254713905292796602797815540, −9.049941186375481284905649155504, −8.263432285350098434822795626642, −6.83373062678915502903504633990, −6.04281915352766135368350069784, −5.13829038734751508547847478604, −4.14380103075318758192360748752, −3.11115739126294715955200221075, −2.15688751935328649074336191333, −0.42041226475841902890997765400, 2.14938501207734666188755677860, 3.51798185261963483454328976472, 4.50501132709917511975588546599, 5.19969124833334176196234113732, 6.43442767147525845598159363448, 6.97812772990587686738503920773, 7.86145985445367830670682974346, 8.663136324695692545484630603182, 9.668777407500002678384191099324, 10.36032031492768836957360477650

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