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Properties

Label 2-30e2-100.23-c1-0-24
Degree $2$
Conductor $900$
Sign $0.998 + 0.0451i$
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  + (−1.38 + 0.262i)2-s + (1.86 − 0.730i)4-s + (−1.64 − 1.51i)5-s + (1.86 + 1.86i)7-s + (−2.39 + 1.50i)8-s + (2.68 + 1.67i)10-s + (1.78 + 2.45i)11-s + (−0.962 − 6.07i)13-s + (−3.07 − 2.09i)14-s + (2.93 − 2.72i)16-s + (0.217 + 0.110i)17-s + (2.00 + 6.17i)19-s + (−4.17 − 1.61i)20-s + (−3.12 − 2.94i)22-s + (0.523 − 3.30i)23-s + ⋯
L(s)  = 1  + (−0.982 + 0.185i)2-s + (0.930 − 0.365i)4-s + (−0.735 − 0.676i)5-s + (0.704 + 0.704i)7-s + (−0.846 + 0.532i)8-s + (0.849 + 0.528i)10-s + (0.537 + 0.739i)11-s + (−0.266 − 1.68i)13-s + (−0.823 − 0.561i)14-s + (0.732 − 0.680i)16-s + (0.0526 + 0.0268i)17-s + (0.460 + 1.41i)19-s + (−0.932 − 0.361i)20-s + (−0.665 − 0.627i)22-s + (0.109 − 0.688i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.998 + 0.0451i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (523, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ 0.998 + 0.0451i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.966271 - 0.0218157i\)
\(L(\frac12)\) \(\approx\) \(0.966271 - 0.0218157i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 - 0.262i)T \)
3 \( 1 \)
5 \( 1 + (1.64 + 1.51i)T \)
good7 \( 1 + (-1.86 - 1.86i)T + 7iT^{2} \)
11 \( 1 + (-1.78 - 2.45i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.962 + 6.07i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-0.217 - 0.110i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-2.00 - 6.17i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-0.523 + 3.30i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (-2.54 - 0.827i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.58 - 0.838i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-4.97 + 0.787i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (1.92 + 1.39i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (0.914 - 0.914i)T - 43iT^{2} \)
47 \( 1 + (-10.2 + 5.22i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-4.36 + 2.22i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (-5.24 - 3.80i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-10.7 + 7.84i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-4.31 + 8.47i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (-2.81 - 0.914i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (6.76 + 1.07i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (2.50 - 7.72i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-6.29 - 3.20i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (-10.2 - 14.0i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-1.60 - 3.15i)T + (-57.0 + 78.4i)T^{2} \)
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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

−10.03050908895239313108867585038, −9.134522851035322842342280892944, −8.248540773912681179183970386644, −7.944276123143210370927112719931, −6.99189272630317674236373177297, −5.68942089884464070319731025395, −5.07899706154149297901166404169, −3.64569919641303120007407370857, −2.23106252435629148023811563569, −0.910070674727749709971003436745, 0.956427664716119760589079567813, 2.42145382921263304647904302546, 3.63499176339953054024332661735, 4.52171399839902748116645647576, 6.17579442263768232579069355161, 7.15276497149678747534552984457, 7.40063292039675397122257441615, 8.563507848424284870647813093308, 9.176059349727462290902855834731, 10.14382577749952698812806513125

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