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Properties

Label 2-30e2-45.32-c1-0-1
Degree $2$
Conductor $900$
Sign $-0.784 + 0.619i$
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  + (−0.866 + 1.5i)3-s + (−0.866 + 3.23i)7-s + (−1.5 − 2.59i)9-s + (−4.09 + 2.36i)11-s + (0.633 + 2.36i)13-s + (3 − 3i)17-s − 6.19i·19-s + (−4.09 − 4.09i)21-s + (−5.59 + 1.5i)23-s + 5.19·27-s + (−2.13 − 3.69i)29-s + (−3.09 + 5.36i)31-s − 8.19i·33-s + (−0.464 − 0.464i)37-s + (−4.09 − 1.09i)39-s + ⋯
L(s)  = 1  + (−0.499 + 0.866i)3-s + (−0.327 + 1.22i)7-s + (−0.5 − 0.866i)9-s + (−1.23 + 0.713i)11-s + (0.175 + 0.656i)13-s + (0.727 − 0.727i)17-s − 1.42i·19-s + (−0.894 − 0.894i)21-s + (−1.16 + 0.312i)23-s + 1.00·27-s + (−0.396 − 0.686i)29-s + (−0.556 + 0.963i)31-s − 1.42i·33-s + (−0.0762 − 0.0762i)37-s + (−0.656 − 0.175i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 + 0.619i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.784 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.784 + 0.619i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ -0.784 + 0.619i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0940274 - 0.270903i\)
\(L(\frac12)\) \(\approx\) \(0.0940274 - 0.270903i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 1.5i)T \)
5 \( 1 \)
good7 \( 1 + (0.866 - 3.23i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (4.09 - 2.36i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.633 - 2.36i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-3 + 3i)T - 17iT^{2} \)
19 \( 1 + 6.19iT - 19T^{2} \)
23 \( 1 + (5.59 - 1.5i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (2.13 + 3.69i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.09 - 5.36i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.464 + 0.464i)T + 37iT^{2} \)
41 \( 1 + (-0.401 - 0.232i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.09 - 1.09i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (12.6 + 3.40i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.19 - 2.19i)T + 53iT^{2} \)
59 \( 1 + (-4.73 + 8.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.59 + 6.23i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (15.0 - 4.03i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 5.66iT - 71T^{2} \)
73 \( 1 + (-2.53 + 2.53i)T - 73iT^{2} \)
79 \( 1 + (1.73 - i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.696 + 2.59i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 14.6T + 89T^{2} \)
97 \( 1 + (3.92 - 14.6i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.55085945435755646317514881020, −9.648225294309594211087042512940, −9.264314287197336182095980471632, −8.252877256240617699986930109128, −7.13949063745174852954338432005, −6.09144972231817143429971926762, −5.28588932728506800272323401808, −4.65282707428201700890840398134, −3.29458958333699274333381213721, −2.29159419901945901890978520465, 0.14181129691223750298595013489, 1.52070063059286995317270522527, 3.05028139069097754528615831160, 4.10375394161474824748314546942, 5.61269686984226667423602853734, 5.93893860173079990619872459182, 7.16802838009861923719497661144, 7.895961092858634095220957381787, 8.302645970500459753756484985611, 10.00035924982027136460418335563

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