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Properties

Label 2-30e2-25.4-c1-0-5
Degree $2$
Conductor $900$
Sign $0.983 + 0.182i$
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  + (−0.892 − 2.05i)5-s + 4.13i·7-s + (1.16 − 3.58i)11-s + (0.664 − 0.215i)13-s + (3.11 + 4.28i)17-s + (4.63 − 3.37i)19-s + (−5.19 − 1.68i)23-s + (−3.40 + 3.66i)25-s + (5.68 + 4.12i)29-s + (8.16 − 5.93i)31-s + (8.47 − 3.69i)35-s + (5.50 − 1.78i)37-s + (2.03 + 6.27i)41-s + 4.79i·43-s + (5.68 − 7.82i)47-s + ⋯
L(s)  = 1  + (−0.399 − 0.916i)5-s + 1.56i·7-s + (0.350 − 1.08i)11-s + (0.184 − 0.0598i)13-s + (0.754 + 1.03i)17-s + (1.06 − 0.773i)19-s + (−1.08 − 0.352i)23-s + (−0.681 + 0.732i)25-s + (1.05 + 0.766i)29-s + (1.46 − 1.06i)31-s + (1.43 − 0.623i)35-s + (0.904 − 0.293i)37-s + (0.318 + 0.979i)41-s + 0.731i·43-s + (0.829 − 1.14i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.983 + 0.182i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ 0.983 + 0.182i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53500 - 0.141447i\)
\(L(\frac12)\) \(\approx\) \(1.53500 - 0.141447i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (0.892 + 2.05i)T \)
good7 \( 1 - 4.13iT - 7T^{2} \)
11 \( 1 + (-1.16 + 3.58i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.664 + 0.215i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-3.11 - 4.28i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-4.63 + 3.37i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (5.19 + 1.68i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (-5.68 - 4.12i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-8.16 + 5.93i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-5.50 + 1.78i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.03 - 6.27i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 4.79iT - 43T^{2} \)
47 \( 1 + (-5.68 + 7.82i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.99 + 2.74i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.230 - 0.708i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.64 + 11.2i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (2.81 + 3.88i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (2.54 + 1.84i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-10.2 - 3.32i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (7.38 + 5.36i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-4.96 - 6.82i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (1.04 - 3.22i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (6.13 - 8.44i)T + (-29.9 - 92.2i)T^{2} \)
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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

−9.853440304357027471890206706080, −9.151290560078966403261101638818, −8.301595817362640221545599535399, −8.036869527479437598230770038301, −6.34821867351093976119513978124, −5.75217322568778384262486539511, −4.89260364683545092762735735256, −3.70238488571697994742963983064, −2.58385232814582266925757994985, −1.02043929866514202951143027263, 1.08221468217501861048076035849, 2.78315049879819323008207963147, 3.86471623556601590441292371907, 4.52065086814849890363090784689, 5.94353823100901633058848009601, 7.04383067538342380599833553453, 7.38048654258028738337003146243, 8.160695106559124305867965692633, 9.783140968012411253874391606192, 10.01210807057488186427773454394

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