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Properties

Label 2-150-15.14-c2-0-7
Degree $2$
Conductor $150$
Sign $0.992 + 0.123i$
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  + 1.41·2-s + (2.82 − i)3-s + 2.00·4-s + (4.00 − 1.41i)6-s + 7i·7-s + 2.82·8-s + (7.00 − 5.65i)9-s + 8.48i·11-s + (5.65 − 2.00i)12-s − 25i·13-s + 9.89i·14-s + 4.00·16-s − 25.4·17-s + (9.89 − 8.00i)18-s + 7·19-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.942 − 0.333i)3-s + 0.500·4-s + (0.666 − 0.235i)6-s + i·7-s + 0.353·8-s + (0.777 − 0.628i)9-s + 0.771i·11-s + (0.471 − 0.166i)12-s − 1.92i·13-s + 0.707i·14-s + 0.250·16-s − 1.49·17-s + (0.549 − 0.444i)18-s + 0.368·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.123i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.992 + 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $0.992 + 0.123i$
Analytic conductor: \(4.08720\)
Root analytic conductor: \(2.02168\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :1),\ 0.992 + 0.123i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.76982 - 0.171686i\)
\(L(\frac12)\) \(\approx\) \(2.76982 - 0.171686i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

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

−12.73037481960288706790995040099, −12.29021305612914621379412695064, −10.81082504500249598821948588992, −9.598915750765400367507950565266, −8.474217127430250598651345437879, −7.49775068565058123540362326641, −6.21612850395450231925346261319, −4.89203463901663889370478820107, −3.29484216429254846579926474801, −2.14134769988902874850174973864, 2.11697732346012028174968242860, 3.83282207342976403828191854889, 4.48277828975855890491453440692, 6.39916664883773626431957529273, 7.41560763429780225776942876713, 8.660554720957281158574951003739, 9.723458454303818083979509166708, 10.88407475059616794955947206421, 11.76542527567509386761947946235, 13.32474903569381141067628413503

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