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Properties

Label 2-30e2-45.29-c2-0-33
Degree $2$
Conductor $900$
Sign $-0.891 + 0.453i$
Analytic cond. $24.5232$
Root an. cond. $4.95209$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.05 − 2.18i)3-s + (7.02 + 4.05i)7-s + (−0.558 − 8.98i)9-s + (−17.6 − 10.1i)11-s + (−5.29 + 3.05i)13-s − 17.9·17-s − 9.11·19-s + (23.3 − 7.02i)21-s + (−16.7 − 29.0i)23-s + (−20.7 − 17.2i)27-s + (−14.4 − 8.31i)29-s + (11.1 + 19.3i)31-s + (−58.4 + 17.6i)33-s − 50.4i·37-s + (−4.19 + 17.8i)39-s + ⋯
L(s)  = 1  + (0.684 − 0.728i)3-s + (1.00 + 0.579i)7-s + (−0.0620 − 0.998i)9-s + (−1.60 − 0.924i)11-s + (−0.407 + 0.235i)13-s − 1.05·17-s − 0.479·19-s + (1.11 − 0.334i)21-s + (−0.729 − 1.26i)23-s + (−0.769 − 0.638i)27-s + (−0.496 − 0.286i)29-s + (0.360 + 0.624i)31-s + (−1.77 + 0.533i)33-s − 1.36i·37-s + (−0.107 + 0.458i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.891 + 0.453i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ -0.891 + 0.453i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.308485557\)
\(L(\frac12)\) \(\approx\) \(1.308485557\)
\(L(2)\) 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 \( 1 + (-2.05 + 2.18i)T \)
5 \( 1 \)
good7 \( 1 + (-7.02 - 4.05i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (17.6 + 10.1i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (5.29 - 3.05i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 17.9T + 289T^{2} \)
19 \( 1 + 9.11T + 361T^{2} \)
23 \( 1 + (16.7 + 29.0i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (14.4 + 8.31i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-11.1 - 19.3i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 50.4iT - 1.36e3T^{2} \)
41 \( 1 + (-29.9 + 17.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-19.9 - 11.5i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-19.1 + 33.1i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 19.0T + 2.80e3T^{2} \)
59 \( 1 + (-2.96 + 1.71i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-23.1 + 40.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (5.45 - 3.14i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 35.9iT - 5.04e3T^{2} \)
73 \( 1 + 47.3iT - 5.32e3T^{2} \)
79 \( 1 + (42.2 - 73.2i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (19.1 - 33.1i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 143. iT - 7.92e3T^{2} \)
97 \( 1 + (69.9 + 40.3i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−9.280203026383895212289013032042, −8.436341597386144988561864160503, −8.118142368142515118200582117488, −7.16314730366650439710208818359, −6.10959369840129620353979584197, −5.21771354058363846593867900219, −4.09096369349225989902332625733, −2.58573248861260044724074813166, −2.13085709399936341597146129830, −0.34802438464279193018051504883, 1.88068127901219124197297961190, 2.77191176902965249641059941206, 4.21966635286610208204017709909, 4.72417869354095726685789343148, 5.64351956929280810480214282796, 7.29431013309404258564795929763, 7.77511781206597397412429161457, 8.488366098351870286820907662483, 9.578590182796411356639760668461, 10.23325230632350067092850614192

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