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Properties

Label 2-30e2-36.23-c1-0-57
Degree $2$
Conductor $900$
Sign $0.472 + 0.881i$
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  + (0.707 + 1.22i)2-s + (−1.72 − 0.178i)3-s + (−0.999 + 1.73i)4-s + (−1 − 2.23i)6-s + (−4.55 + 2.62i)7-s − 2.82·8-s + (2.93 + 0.614i)9-s + (2.03 − 2.80i)12-s + (−6.43 − 3.71i)14-s + (−2.00 − 3.46i)16-s + (1.32 + 4.03i)18-s + (8.30 − 3.71i)21-s + (3.75 − 6.50i)23-s + (4.87 + 0.504i)24-s + (−4.94 − 1.58i)27-s − 10.5i·28-s + ⋯
L(s)  = 1  + (0.499 + 0.866i)2-s + (−0.994 − 0.102i)3-s + (−0.499 + 0.866i)4-s + (−0.408 − 0.912i)6-s + (−1.72 + 0.993i)7-s − 0.999·8-s + (0.978 + 0.204i)9-s + (0.586 − 0.809i)12-s + (−1.72 − 0.993i)14-s + (−0.500 − 0.866i)16-s + (0.312 + 0.950i)18-s + (1.81 − 0.810i)21-s + (0.782 − 1.35i)23-s + (0.994 + 0.102i)24-s + (−0.952 − 0.304i)27-s − 1.98i·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.472 + 0.881i$
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: \((2,\ 900,\ (\ :1/2),\ 0.472 + 0.881i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.153469 - 0.0918682i\)
\(L(\frac12)\) \(\approx\) \(0.153469 - 0.0918682i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 1.22i)T \)
3 \( 1 + (1.72 + 0.178i)T \)
5 \( 1 \)
good7 \( 1 + (4.55 - 2.62i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (-3.75 + 6.50i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.627 + 0.362i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + (7.06 + 4.07i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.73 - 1.58i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.63 + 2.82i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.80 + 6.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.66 + 2.11i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.99 - 13.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 14.1iT - 89T^{2} \)
97 \( 1 + (-48.5 + 84.0i)T^{2} \)
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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

−9.814172402039930275954467111036, −9.129760194212501336196918689841, −8.185748948616300303741747424409, −6.76541964500077942370382731856, −6.65114265313599219802753951654, −5.70320628538707107853486446881, −4.98531880601555883559220476175, −3.76745704118974383211617248403, −2.67579819163448352449985853073, −0.089811104387721747915361030867, 1.23609300387641545432217469769, 3.11313869711194775571115302973, 3.83784539189072204881172048793, 4.84302243808876791991209673299, 5.86338302979289170312740671385, 6.58449164247589100780621406627, 7.35992221875822544437328554575, 9.092123213350359210724643521705, 9.815758415683030983329215816337, 10.29873171665352692739037775882

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