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Properties

Label 2-150-3.2-c2-0-0
Degree $2$
Conductor $150$
Sign $-0.942 - 0.333i$
Analytic cond. $4.08720$
Root an. cond. $2.02168$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s + (−1 + 2.82i)3-s − 2.00·4-s + (4.00 + 1.41i)6-s − 7·7-s + 2.82i·8-s + (−7.00 − 5.65i)9-s − 8.48i·11-s + (2.00 − 5.65i)12-s − 25·13-s + 9.89i·14-s + 4.00·16-s + 25.4i·17-s + (−8.00 + 9.89i)18-s − 7·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.333 + 0.942i)3-s − 0.500·4-s + (0.666 + 0.235i)6-s − 7-s + 0.353i·8-s + (−0.777 − 0.628i)9-s − 0.771i·11-s + (0.166 − 0.471i)12-s − 1.92·13-s + 0.707i·14-s + 0.250·16-s + 1.49i·17-s + (−0.444 + 0.549i)18-s − 0.368·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 - 0.333i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.942 - 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $-0.942 - 0.333i$
Analytic conductor: \(4.08720\)
Root analytic conductor: \(2.02168\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :1),\ -0.942 - 0.333i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0150851 + 0.0879225i\)
\(L(\frac12)\) \(\approx\) \(0.0150851 + 0.0879225i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 + (1 - 2.82i)T \)
5 \( 1 \)
good7 \( 1 + 7T + 49T^{2} \)
11 \( 1 + 8.48iT - 121T^{2} \)
13 \( 1 + 25T + 169T^{2} \)
17 \( 1 - 25.4iT - 289T^{2} \)
19 \( 1 + 7T + 361T^{2} \)
23 \( 1 + 25.4iT - 529T^{2} \)
29 \( 1 - 42.4iT - 841T^{2} \)
31 \( 1 + 7T + 961T^{2} \)
37 \( 1 - 2T + 1.36e3T^{2} \)
41 \( 1 - 8.48iT - 1.68e3T^{2} \)
43 \( 1 - 41T + 1.84e3T^{2} \)
47 \( 1 - 2.20e3T^{2} \)
53 \( 1 + 59.3iT - 2.80e3T^{2} \)
59 \( 1 + 33.9iT - 3.48e3T^{2} \)
61 \( 1 + T + 3.72e3T^{2} \)
67 \( 1 - 17T + 4.48e3T^{2} \)
71 \( 1 - 42.4iT - 5.04e3T^{2} \)
73 \( 1 + 70T + 5.32e3T^{2} \)
79 \( 1 + 58T + 6.24e3T^{2} \)
83 \( 1 - 118. iT - 6.88e3T^{2} \)
89 \( 1 - 135. iT - 7.92e3T^{2} \)
97 \( 1 + 49T + 9.40e3T^{2} \)
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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

−12.80077935746623139225641811608, −12.30583293590574728480858659903, −10.98677574082474980001972357672, −10.26038770582260392884633175811, −9.465384406386950625864871437334, −8.426282351742969971479270964028, −6.58848367226181901897090713325, −5.32220201292438247460489817266, −4.03583413641834196898188828966, −2.82007134039990397847283642084, 0.05563988649986282565162414076, 2.59162756444145156550241095132, 4.76263833630216589244230829677, 5.94939396411308742126382586116, 7.19719697374105218027772967004, 7.51349113000620499237057728998, 9.256703092164790773438820338651, 9.995475528195160896285160164942, 11.71425854489656058054119170869, 12.45173475548742750593217424520

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