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Properties

Label 2-30e2-25.4-c1-0-10
Degree $2$
Conductor $900$
Sign $-0.260 + 0.965i$
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  + (0.900 − 2.04i)5-s − 0.957i·7-s + (1.67 − 5.15i)11-s + (−1.92 + 0.625i)13-s + (−0.377 − 0.520i)17-s + (−4.07 + 2.96i)19-s + (3.34 + 1.08i)23-s + (−3.37 − 3.68i)25-s + (−8.20 − 5.96i)29-s + (−2.98 + 2.16i)31-s + (−1.95 − 0.862i)35-s + (10.7 − 3.49i)37-s + (−1.08 − 3.35i)41-s − 0.766i·43-s + (−2.90 + 3.99i)47-s + ⋯
L(s)  = 1  + (0.402 − 0.915i)5-s − 0.361i·7-s + (0.504 − 1.55i)11-s + (−0.533 + 0.173i)13-s + (−0.0916 − 0.126i)17-s + (−0.935 + 0.679i)19-s + (0.697 + 0.226i)23-s + (−0.675 − 0.737i)25-s + (−1.52 − 1.10i)29-s + (−0.536 + 0.389i)31-s + (−0.331 − 0.145i)35-s + (1.76 − 0.574i)37-s + (−0.170 − 0.523i)41-s − 0.116i·43-s + (−0.423 + 0.582i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.260 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.260 + 0.965i$
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: \((2,\ 900,\ (\ :1/2),\ -0.260 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.857287 - 1.11892i\)
\(L(\frac12)\) \(\approx\) \(0.857287 - 1.11892i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.900 + 2.04i)T \)
good7 \( 1 + 0.957iT - 7T^{2} \)
11 \( 1 + (-1.67 + 5.15i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (1.92 - 0.625i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.377 + 0.520i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (4.07 - 2.96i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-3.34 - 1.08i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (8.20 + 5.96i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.98 - 2.16i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-10.7 + 3.49i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.08 + 3.35i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 0.766iT - 43T^{2} \)
47 \( 1 + (2.90 - 3.99i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-3.49 + 4.81i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.45 - 4.48i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.34 + 4.13i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (5.59 + 7.70i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (9.66 + 7.02i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-5.16 - 1.67i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-9.58 - 6.96i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.819 - 1.12i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (0.527 - 1.62i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-8.57 + 11.8i)T + (-29.9 - 92.2i)T^{2} \)
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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

−9.676621148738572660345683157975, −9.063487410266323952301650314527, −8.308082859359928555509412050514, −7.41928698696924112533830799728, −6.18034463852620438527980905628, −5.62287501650849358893457109689, −4.45954219994703082594277516912, −3.57779741434497465619353363964, −2.06239240905996916061167089355, −0.66014306989786019585934131675, 1.89089935889164097954452842383, 2.75094304786042161415297091172, 4.08963977467431385930125774413, 5.07765921397335518758545251750, 6.16722492088966971620585030130, 7.01148621585875371410537329216, 7.53648071728691354404891859511, 8.912427241660201674354243509490, 9.544230849800483800478750220691, 10.30628895024283440840660235722

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