Properties

Label 2-150-15.14-c10-0-10
Degree $2$
Conductor $150$
Sign $-0.857 + 0.515i$
Analytic cond. $95.3035$
Root an. cond. $9.76235$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 22.6·2-s + (18.8 + 242. i)3-s + 512.·4-s + (−426. − 5.48e3i)6-s − 670. i·7-s − 1.15e4·8-s + (−5.83e4 + 9.12e3i)9-s + 2.33e5i·11-s + (9.64e3 + 1.24e5i)12-s + 3.07e5i·13-s + 1.51e4i·14-s + 2.62e5·16-s + 6.72e5·17-s + (1.32e6 − 2.06e5i)18-s + 1.55e6·19-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.0775 + 0.996i)3-s + 0.500·4-s + (−0.0548 − 0.704i)6-s − 0.0398i·7-s − 0.353·8-s + (−0.987 + 0.154i)9-s + 1.44i·11-s + (0.0387 + 0.498i)12-s + 0.828i·13-s + 0.0282i·14-s + 0.250·16-s + 0.473·17-s + (0.698 − 0.109i)18-s + 0.626·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 + 0.515i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.857 + 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $-0.857 + 0.515i$
Analytic conductor: \(95.3035\)
Root analytic conductor: \(9.76235\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :5),\ -0.857 + 0.515i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.9329129180\)
\(L(\frac12)\) \(\approx\) \(0.9329129180\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 22.6T \)
3 \( 1 + (-18.8 - 242. i)T \)
5 \( 1 \)
good7 \( 1 + 670. iT - 2.82e8T^{2} \)
11 \( 1 - 2.33e5iT - 2.59e10T^{2} \)
13 \( 1 - 3.07e5iT - 1.37e11T^{2} \)
17 \( 1 - 6.72e5T + 2.01e12T^{2} \)
19 \( 1 - 1.55e6T + 6.13e12T^{2} \)
23 \( 1 + 5.57e6T + 4.14e13T^{2} \)
29 \( 1 - 2.97e7iT - 4.20e14T^{2} \)
31 \( 1 - 3.09e7T + 8.19e14T^{2} \)
37 \( 1 - 8.56e7iT - 4.80e15T^{2} \)
41 \( 1 - 3.59e7iT - 1.34e16T^{2} \)
43 \( 1 + 3.66e7iT - 2.16e16T^{2} \)
47 \( 1 - 3.28e7T + 5.25e16T^{2} \)
53 \( 1 + 4.59e8T + 1.74e17T^{2} \)
59 \( 1 - 4.88e8iT - 5.11e17T^{2} \)
61 \( 1 + 6.12e7T + 7.13e17T^{2} \)
67 \( 1 - 6.70e8iT - 1.82e18T^{2} \)
71 \( 1 - 1.23e9iT - 3.25e18T^{2} \)
73 \( 1 - 1.08e9iT - 4.29e18T^{2} \)
79 \( 1 - 1.86e9T + 9.46e18T^{2} \)
83 \( 1 - 1.09e9T + 1.55e19T^{2} \)
89 \( 1 + 5.19e9iT - 3.11e19T^{2} \)
97 \( 1 - 1.07e10iT - 7.37e19T^{2} \)
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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

−11.55695950157271927552775589416, −10.32043969183038222716738511269, −9.776276183101144377362020869993, −8.881790798236681202262144099652, −7.73970606344221288838860321862, −6.59589449175288498047499661534, −5.16140772487430150389119930169, −4.11170392903757222581133334658, −2.75333119558039016847702495297, −1.44982910169287158763740065236, 0.30201616462484960846975386877, 0.988602187253098440295486836816, 2.38692124945801935214171434196, 3.42446971051215239225600529119, 5.61171239955262146575405816556, 6.31748778654170071500278733496, 7.73178818196458479851073832176, 8.183694796573246844881843276332, 9.330057435640598460191562345860, 10.57596540499924111605082313752

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