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Properties

Label 2-30e2-180.7-c1-0-30
Degree $2$
Conductor $900$
Sign $0.865 - 0.501i$
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  + (0.366 + 1.36i)2-s + (−1.34 − 1.09i)3-s + (−1.73 + i)4-s + (1 − 2.23i)6-s + (−0.159 − 0.596i)7-s + (−2 − 1.99i)8-s + (0.614 + 2.93i)9-s + (3.42 + 0.547i)12-s + (0.756 − 0.436i)14-s + (1.99 − 3.46i)16-s + (−3.78 + 1.91i)18-s + (−0.436 + 0.976i)21-s + (1.94 − 7.25i)23-s + (0.504 + 4.87i)24-s + (2.38 − 4.61i)27-s + (0.872 + 0.872i)28-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.776 − 0.630i)3-s + (−0.866 + 0.5i)4-s + (0.408 − 0.912i)6-s + (−0.0603 − 0.225i)7-s + (−0.707 − 0.707i)8-s + (0.204 + 0.978i)9-s + (0.987 + 0.158i)12-s + (0.202 − 0.116i)14-s + (0.499 − 0.866i)16-s + (−0.892 + 0.451i)18-s + (−0.0952 + 0.212i)21-s + (0.405 − 1.51i)23-s + (0.102 + 0.994i)24-s + (0.458 − 0.888i)27-s + (0.164 + 0.164i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 - 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.865 - 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.865 - 0.501i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ 0.865 - 0.501i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12231 + 0.301699i\)
\(L(\frac12)\) \(\approx\) \(1.12231 + 0.301699i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 - 1.36i)T \)
3 \( 1 + (1.34 + 1.09i)T \)
5 \( 1 \)
good7 \( 1 + (0.159 + 0.596i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (11.2 + 6.5i)T^{2} \)
17 \( 1 + 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (-1.94 + 7.25i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-9.30 - 5.37i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 37iT^{2} \)
41 \( 1 + (-4.93 - 8.55i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-12.2 + 3.29i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-0.845 - 3.15i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.80 + 6.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (15.2 + 4.09i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-15.4 + 4.13i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 - 12.4iT - 89T^{2} \)
97 \( 1 + (84.0 - 48.5i)T^{2} \)
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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

−10.27301081576774888739798819681, −9.092415042080919362048847967121, −8.251068284713785471337668779171, −7.43686698030334775555742741871, −6.65083568090252135567998731557, −6.06797743672862698952617258815, −5.00790140392886092950637385626, −4.31811775435281431876398034600, −2.76917568892629357273073955292, −0.812367582420370736906836009068, 0.951725980067066834842713159446, 2.60467806604339243167815839685, 3.76229442734433123132490453393, 4.56544269354751431107806880015, 5.52393448759633320835763977329, 6.16419236855690564818103774551, 7.50618666855047920969616353993, 8.843687021576607653854709632061, 9.398194833153555132529908372992, 10.27573166069011446398841317288

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