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Properties

Label 2-150-5.4-c9-0-22
Degree $2$
Conductor $150$
Sign $-0.447 + 0.894i$
Analytic cond. $77.2553$
Root an. cond. $8.78950$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 16i·2-s + 81i·3-s − 256·4-s − 1.29e3·6-s − 1.03e4i·7-s − 4.09e3i·8-s − 6.56e3·9-s + 2.74e4·11-s − 2.07e4i·12-s + 1.69e5i·13-s + 1.65e5·14-s + 6.55e4·16-s − 3.85e5i·17-s − 1.04e5i·18-s + 6.37e5·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s − 1.62i·7-s − 0.353i·8-s − 0.333·9-s + 0.564·11-s − 0.288i·12-s + 1.64i·13-s + 1.15·14-s + 0.250·16-s − 1.11i·17-s − 0.235i·18-s + 1.12·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(77.2553\)
Root analytic conductor: \(8.78950\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :9/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.1071349486\)
\(L(\frac12)\) \(\approx\) \(0.1071349486\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 16iT \)
3 \( 1 - 81iT \)
5 \( 1 \)
good7 \( 1 + 1.03e4iT - 4.03e7T^{2} \)
11 \( 1 - 2.74e4T + 2.35e9T^{2} \)
13 \( 1 - 1.69e5iT - 1.06e10T^{2} \)
17 \( 1 + 3.85e5iT - 1.18e11T^{2} \)
19 \( 1 - 6.37e5T + 3.22e11T^{2} \)
23 \( 1 - 1.29e6iT - 1.80e12T^{2} \)
29 \( 1 + 7.16e6T + 1.45e13T^{2} \)
31 \( 1 + 7.03e6T + 2.64e13T^{2} \)
37 \( 1 - 1.92e6iT - 1.29e14T^{2} \)
41 \( 1 - 8.89e6T + 3.27e14T^{2} \)
43 \( 1 + 3.24e7iT - 5.02e14T^{2} \)
47 \( 1 - 1.72e7iT - 1.11e15T^{2} \)
53 \( 1 - 2.06e7iT - 3.29e15T^{2} \)
59 \( 1 - 6.31e7T + 8.66e15T^{2} \)
61 \( 1 + 6.37e7T + 1.16e16T^{2} \)
67 \( 1 - 1.45e8iT - 2.72e16T^{2} \)
71 \( 1 + 3.67e8T + 4.58e16T^{2} \)
73 \( 1 + 2.52e8iT - 5.88e16T^{2} \)
79 \( 1 - 1.85e8T + 1.19e17T^{2} \)
83 \( 1 - 4.67e8iT - 1.86e17T^{2} \)
89 \( 1 + 5.79e8T + 3.50e17T^{2} \)
97 \( 1 + 1.31e9iT - 7.60e17T^{2} \)
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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

−10.96009275492370653158665904142, −9.603253703867914971233516560787, −9.187551005112268816817851132709, −7.41745456307806491320327149514, −7.05070671152742394285769224970, −5.50837456731519403835867775203, −4.29368150312837310644821235200, −3.61071955435856050527414983590, −1.41772467153485289930298057011, −0.02552209134343803728144542007, 1.39853929545142203785052178521, 2.47647336718894713162788342342, 3.51187431859684244072130794673, 5.34644243969763151651111104780, 6.00370517962808350599365649527, 7.70138190425169512607490376527, 8.652962542885684130116887611658, 9.521573456783785828346555635077, 10.81961445471590200191409482081, 11.74911714832832061663685457321

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