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Properties

Label 2-150-15.14-c2-0-6
Degree $2$
Conductor $150$
Sign $0.612 + 0.790i$
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 + (2.94 − 0.581i)3-s + 2.00·4-s + (−4.16 + 0.821i)6-s − 11.4i·7-s − 2.82·8-s + (8.32 − 3.42i)9-s + 8.48i·11-s + (5.88 − 1.16i)12-s − 10i·13-s + 16.2i·14-s + 4.00·16-s − 3.55·17-s + (−11.7 + 4.83i)18-s + 10.9·19-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.981 − 0.193i)3-s + 0.500·4-s + (−0.693 + 0.136i)6-s − 1.64i·7-s − 0.353·8-s + (0.924 − 0.380i)9-s + 0.771i·11-s + (0.490 − 0.0968i)12-s − 0.769i·13-s + 1.16i·14-s + 0.250·16-s − 0.209·17-s + (−0.654 + 0.268i)18-s + 0.577·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $0.612 + 0.790i$
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: \((2,\ 150,\ (\ :1),\ 0.612 + 0.790i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.29738 - 0.636503i\)
\(L(\frac12)\) \(\approx\) \(1.29738 - 0.636503i\)
\(L(2)\) 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 \( 1 + (-2.94 + 0.581i)T \)
5 \( 1 \)
good7 \( 1 + 11.4iT - 49T^{2} \)
11 \( 1 - 8.48iT - 121T^{2} \)
13 \( 1 + 10iT - 169T^{2} \)
17 \( 1 + 3.55T + 289T^{2} \)
19 \( 1 - 10.9T + 361T^{2} \)
23 \( 1 - 17.6T + 529T^{2} \)
29 \( 1 + 26.8iT - 841T^{2} \)
31 \( 1 - 8T + 961T^{2} \)
37 \( 1 - 59.9iT - 1.36e3T^{2} \)
41 \( 1 - 20.5iT - 1.68e3T^{2} \)
43 \( 1 - 42.4iT - 1.84e3T^{2} \)
47 \( 1 + 88.2T + 2.20e3T^{2} \)
53 \( 1 - 3.55T + 2.80e3T^{2} \)
59 \( 1 - 77.7iT - 3.48e3T^{2} \)
61 \( 1 - 21.9T + 3.72e3T^{2} \)
67 \( 1 - 53.5iT - 4.48e3T^{2} \)
71 \( 1 - 69.2iT - 5.04e3T^{2} \)
73 \( 1 - 12.0iT - 5.32e3T^{2} \)
79 \( 1 - 9.02T + 6.24e3T^{2} \)
83 \( 1 - 0.688T + 6.88e3T^{2} \)
89 \( 1 + 7.10iT - 7.92e3T^{2} \)
97 \( 1 + 111. iT - 9.40e3T^{2} \)
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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

−12.90386915706135992980779136647, −11.44239530843829819821358817443, −10.16712581389159193903628664398, −9.753364748065543456912614518034, −8.302019625835823375154443319317, −7.52224832058359312724795437770, −6.72076123412124496393607688323, −4.48348761851454429059692022461, −3.06746458144341803757100989042, −1.19604009509123711789060283645, 2.04113190669407753718236919261, 3.28730966927018693150170456063, 5.26164333761661299026295192674, 6.70480651516010999897889805726, 8.056002684430771424687110508271, 8.984666996717253653216729402224, 9.342570467536504401871306386065, 10.79611197785232050911931946165, 11.84304648807731058971515069601, 12.86344062133253555660029906318

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