↑

Properties

Label 2-150-15.2-c7-0-31
Degree $2$
Conductor $150$
Sign $0.965 + 0.259i$
Analytic cond. $46.8577$
Root an. cond. $6.84527$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−5.65 + 5.65i)2-s + (46.6 − 3.26i)3-s − 64.0i·4-s + (−245. + 282. i)6-s + (918. + 918. i)7-s + (362. + 362. i)8-s + (2.16e3 − 304. i)9-s − 7.13e3i·11-s + (−209. − 2.98e3i)12-s + (1.04e4 − 1.04e4i)13-s − 1.03e4·14-s − 4.09e3·16-s + (−1.98e4 + 1.98e4i)17-s + (−1.05e4 + 1.39e4i)18-s − 4.24e4i·19-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.997 − 0.0698i)3-s − 0.500i·4-s + (−0.463 + 0.533i)6-s + (1.01 + 1.01i)7-s + (0.250 + 0.250i)8-s + (0.990 − 0.139i)9-s − 1.61i·11-s + (−0.0349 − 0.498i)12-s + (1.31 − 1.31i)13-s − 1.01·14-s − 0.250·16-s + (−0.980 + 0.980i)17-s + (−0.425 + 0.564i)18-s − 1.41i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.259i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.965 + 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $0.965 + 0.259i$
Analytic conductor: \(46.8577\)
Root analytic conductor: \(6.84527\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :7/2),\ 0.965 + 0.259i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.726520127\)
\(L(\frac12)\) \(\approx\) \(2.726520127\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.65 - 5.65i)T \)
3 \( 1 + (-46.6 + 3.26i)T \)
5 \( 1 \)
good7 \( 1 + (-918. - 918. i)T + 8.23e5iT^{2} \)
11 \( 1 + 7.13e3iT - 1.94e7T^{2} \)
13 \( 1 + (-1.04e4 + 1.04e4i)T - 6.27e7iT^{2} \)
17 \( 1 + (1.98e4 - 1.98e4i)T - 4.10e8iT^{2} \)
19 \( 1 + 4.24e4iT - 8.93e8T^{2} \)
23 \( 1 + (-4.59e3 - 4.59e3i)T + 3.40e9iT^{2} \)
29 \( 1 + 1.04e5T + 1.72e10T^{2} \)
31 \( 1 + 3.87e4T + 2.75e10T^{2} \)
37 \( 1 + (1.24e5 + 1.24e5i)T + 9.49e10iT^{2} \)
41 \( 1 - 3.28e5iT - 1.94e11T^{2} \)
43 \( 1 + (-1.23e5 + 1.23e5i)T - 2.71e11iT^{2} \)
47 \( 1 + (-6.31e5 + 6.31e5i)T - 5.06e11iT^{2} \)
53 \( 1 + (7.06e5 + 7.06e5i)T + 1.17e12iT^{2} \)
59 \( 1 - 4.01e5T + 2.48e12T^{2} \)
61 \( 1 - 2.71e6T + 3.14e12T^{2} \)
67 \( 1 + (1.30e6 + 1.30e6i)T + 6.06e12iT^{2} \)
71 \( 1 - 1.05e6iT - 9.09e12T^{2} \)
73 \( 1 + (-2.39e6 + 2.39e6i)T - 1.10e13iT^{2} \)
79 \( 1 + 2.40e6iT - 1.92e13T^{2} \)
83 \( 1 + (-3.09e6 - 3.09e6i)T + 2.71e13iT^{2} \)
89 \( 1 - 2.35e6T + 4.42e13T^{2} \)
97 \( 1 + (-7.53e6 - 7.53e6i)T + 8.07e13iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.30104202073015554227164082958, −10.68224163035526211699234628680, −8.930024808059814535099553513752, −8.632601994762727625656300943724, −7.87733240912223576202939818848, −6.32624213009055298751949693321, −5.27817586366752554116072539014, −3.55395797136896938291135450322, −2.19676632306430107644387296615, −0.832268629222879466325752514299, 1.38303109028802929021532996670, 2.05883264885694853108446902009, 3.89407619009794801079697357880, 4.50188297721779130588643918873, 6.94465746869512391311094022825, 7.65231355701250160961608560860, 8.748482784331884450250369400841, 9.614754750311767792652973779124, 10.61043673567060272558749937414, 11.54126763057297224050749813932

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