Properties

Label 2-150-25.12-c2-0-2
Degree $2$
Conductor $150$
Sign $0.446 - 0.894i$
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  + (0.221 + 1.39i)2-s + (−1.54 − 0.786i)3-s + (−1.90 + 0.618i)4-s + (−1.48 − 4.77i)5-s + (0.756 − 2.32i)6-s + (9.35 + 9.35i)7-s + (−1.28 − 2.52i)8-s + (1.76 + 2.42i)9-s + (6.33 − 3.13i)10-s + (13.3 + 9.71i)11-s + (3.42 + 0.541i)12-s + (−1.86 + 11.7i)13-s + (−11.0 + 15.1i)14-s + (−1.45 + 8.53i)15-s + (3.23 − 2.35i)16-s + (17.3 − 8.86i)17-s + ⋯
L(s)  = 1  + (0.110 + 0.698i)2-s + (−0.514 − 0.262i)3-s + (−0.475 + 0.154i)4-s + (−0.297 − 0.954i)5-s + (0.126 − 0.388i)6-s + (1.33 + 1.33i)7-s + (−0.160 − 0.315i)8-s + (0.195 + 0.269i)9-s + (0.633 − 0.313i)10-s + (1.21 + 0.882i)11-s + (0.285 + 0.0451i)12-s + (−0.143 + 0.904i)13-s + (−0.785 + 1.08i)14-s + (−0.0970 + 0.569i)15-s + (0.202 − 0.146i)16-s + (1.02 − 0.521i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.446 - 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.446 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $0.446 - 0.894i$
Analytic conductor: \(4.08720\)
Root analytic conductor: \(2.02168\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :1),\ 0.446 - 0.894i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.16638 + 0.721136i\)
\(L(\frac12)\) \(\approx\) \(1.16638 + 0.721136i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.221 - 1.39i)T \)
3 \( 1 + (1.54 + 0.786i)T \)
5 \( 1 + (1.48 + 4.77i)T \)
good7 \( 1 + (-9.35 - 9.35i)T + 49iT^{2} \)
11 \( 1 + (-13.3 - 9.71i)T + (37.3 + 115. i)T^{2} \)
13 \( 1 + (1.86 - 11.7i)T + (-160. - 52.2i)T^{2} \)
17 \( 1 + (-17.3 + 8.86i)T + (169. - 233. i)T^{2} \)
19 \( 1 + (16.4 + 5.33i)T + (292. + 212. i)T^{2} \)
23 \( 1 + (-17.1 + 2.70i)T + (503. - 163. i)T^{2} \)
29 \( 1 + (-7.90 + 2.56i)T + (680. - 494. i)T^{2} \)
31 \( 1 + (0.527 - 1.62i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (-26.7 - 4.23i)T + (1.30e3 + 423. i)T^{2} \)
41 \( 1 + (41.3 - 30.0i)T + (519. - 1.59e3i)T^{2} \)
43 \( 1 + (-16.8 + 16.8i)T - 1.84e3iT^{2} \)
47 \( 1 + (-9.38 + 18.4i)T + (-1.29e3 - 1.78e3i)T^{2} \)
53 \( 1 + (29.4 + 15.0i)T + (1.65e3 + 2.27e3i)T^{2} \)
59 \( 1 + (49.9 + 68.7i)T + (-1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (62.4 + 45.3i)T + (1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 + (-49.3 + 25.1i)T + (2.63e3 - 3.63e3i)T^{2} \)
71 \( 1 + (-3.44 - 10.5i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (-15.9 + 2.53i)T + (5.06e3 - 1.64e3i)T^{2} \)
79 \( 1 + (84.5 - 27.4i)T + (5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (48.3 + 94.7i)T + (-4.04e3 + 5.57e3i)T^{2} \)
89 \( 1 + (-56.1 + 77.2i)T + (-2.44e3 - 7.53e3i)T^{2} \)
97 \( 1 + (49.5 - 97.2i)T + (-5.53e3 - 7.61e3i)T^{2} \)
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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

−12.70885099888749914551655054429, −12.00443864206374119993048451911, −11.44941190402297175724463695772, −9.458462011261934658716463241792, −8.724055073441618010744440453099, −7.70862431797123477187197183344, −6.42575059875742853489350880929, −5.12944808070291918160087153502, −4.50532475497376474874251078759, −1.63569034644040435868814869524, 1.12033364439302020814868321066, 3.44816181422093227118181956555, 4.39807060039499776560482186495, 5.93535036202325586136288634359, 7.33310445905382213291937995956, 8.398256450193083943188375106249, 10.07853553803058478032539738478, 10.82626583712532099917432839617, 11.26715362065785449837983592947, 12.30904043927069567084683723470

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