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Properties

Label 2-30e2-180.119-c1-0-1
Degree $2$
Conductor $900$
Sign $-0.819 + 0.573i$
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 + 0.295i)2-s + (−1.64 − 0.531i)3-s + (1.82 − 0.817i)4-s + (2.43 + 0.247i)6-s + (−2.08 + 3.61i)7-s + (−2.28 + 1.67i)8-s + (2.43 + 1.75i)9-s + (−3.22 + 5.57i)11-s + (−3.44 + 0.378i)12-s + (1.53 − 0.887i)13-s + (1.81 − 5.61i)14-s + (2.66 − 2.98i)16-s + 0.950·17-s + (−3.88 − 1.70i)18-s − 2.19i·19-s + ⋯
L(s)  = 1  + (−0.977 + 0.209i)2-s + (−0.951 − 0.306i)3-s + (0.912 − 0.408i)4-s + (0.994 + 0.101i)6-s + (−0.788 + 1.36i)7-s + (−0.806 + 0.590i)8-s + (0.811 + 0.584i)9-s + (−0.971 + 1.68i)11-s + (−0.994 + 0.109i)12-s + (0.426 − 0.246i)13-s + (0.485 − 1.49i)14-s + (0.665 − 0.746i)16-s + 0.230·17-s + (−0.915 − 0.401i)18-s − 0.503i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.819 + 0.573i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ -0.819 + 0.573i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0227332 - 0.0721774i\)
\(L(\frac12)\) \(\approx\) \(0.0227332 - 0.0721774i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 - 0.295i)T \)
3 \( 1 + (1.64 + 0.531i)T \)
5 \( 1 \)
good7 \( 1 + (2.08 - 3.61i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.22 - 5.57i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.53 + 0.887i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 0.950T + 17T^{2} \)
19 \( 1 + 2.19iT - 19T^{2} \)
23 \( 1 + (-1.74 + 1.01i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.65 + 3.26i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.46 + 1.42i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.83iT - 37T^{2} \)
41 \( 1 + (6.16 - 3.55i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.65 + 4.59i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.349 + 0.201i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 4.87T + 53T^{2} \)
59 \( 1 + (-1.34 - 2.32i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.04 - 3.54i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.44 + 9.43i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.23T + 71T^{2} \)
73 \( 1 + 1.74iT - 73T^{2} \)
79 \( 1 + (0.214 + 0.123i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.1 + 5.83i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 6.13iT - 89T^{2} \)
97 \( 1 + (-15.7 - 9.08i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.36265773578019029178792621409, −9.853252984327107190919732442079, −9.064395312540465964366297951978, −7.994905009010870658175189302232, −7.21731104582358157175839459894, −6.39553639295459336010379023433, −5.64950169595843359210450618847, −4.83161989047987755089299683268, −2.80169313632431834242878374529, −1.83485039388050321110250940379, 0.06288348438562582030974291662, 1.14427857683208484313900950333, 3.20596671888702095242000964035, 3.87759972531138006674528060837, 5.48396599439314403067804666702, 6.26880400088140472775013522000, 7.09007382237209152218451003489, 7.88839247246327611809660087927, 8.935646905849912559751750184383, 9.830893479801243539900519193255

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