Properties

Label 2-150-15.8-c1-0-5
Degree $2$
Conductor $150$
Sign $0.991 + 0.130i$
Analytic cond. $1.19775$
Root an. cond. $1.09442$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)2-s + (0.448 − 1.67i)3-s + 1.00i·4-s + (1.5 − 0.866i)6-s + (2.44 − 2.44i)7-s + (−0.707 + 0.707i)8-s + (−2.59 − 1.50i)9-s + 5.19i·11-s + (1.67 + 0.448i)12-s + 3.46·14-s − 1.00·16-s + (−2.12 − 2.12i)17-s + (−0.776 − 2.89i)18-s + i·19-s + (−3 − 5.19i)21-s + (−3.67 + 3.67i)22-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.258 − 0.965i)3-s + 0.500i·4-s + (0.612 − 0.353i)6-s + (0.925 − 0.925i)7-s + (−0.250 + 0.250i)8-s + (−0.866 − 0.5i)9-s + 1.56i·11-s + (0.482 + 0.129i)12-s + 0.925·14-s − 0.250·16-s + (−0.514 − 0.514i)17-s + (−0.183 − 0.683i)18-s + 0.229i·19-s + (−0.654 − 1.13i)21-s + (−0.783 + 0.783i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $0.991 + 0.130i$
Analytic conductor: \(1.19775\)
Root analytic conductor: \(1.09442\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :1/2),\ 0.991 + 0.130i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53972 - 0.100729i\)
\(L(\frac12)\) \(\approx\) \(1.53972 - 0.100729i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (-0.448 + 1.67i)T \)
5 \( 1 \)
good7 \( 1 + (-2.44 + 2.44i)T - 7iT^{2} \)
11 \( 1 - 5.19iT - 11T^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 + (2.12 + 2.12i)T + 17iT^{2} \)
19 \( 1 - iT - 19T^{2} \)
23 \( 1 + (4.24 - 4.24i)T - 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + (2.44 - 2.44i)T - 37iT^{2} \)
41 \( 1 + 5.19iT - 41T^{2} \)
43 \( 1 + (2.44 + 2.44i)T + 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 + (-4.24 + 4.24i)T - 53iT^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 + (-3.67 + 3.67i)T - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (6.12 + 6.12i)T + 73iT^{2} \)
79 \( 1 - 14iT - 79T^{2} \)
83 \( 1 + (-2.12 + 2.12i)T - 83iT^{2} \)
89 \( 1 + 15.5T + 89T^{2} \)
97 \( 1 + (-4.89 + 4.89i)T - 97iT^{2} \)
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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

−13.17192093826113317992082671663, −12.19016416029915654248787947267, −11.34725784440517683477289839929, −9.882036399215379994954478530424, −8.425235377073932184631894571370, −7.42801791092927869062854654229, −6.90561952511872856444267621648, −5.29478349274904067171114416408, −4.00531687195082858702010787900, −1.96713013012517432355421949032, 2.47033492955680976088709044635, 3.87838820991488060311248225288, 5.14095023665045280523795529715, 6.05292294595787903370532722539, 8.318738013193146575953123488860, 8.840530825379909451702300437125, 10.21795996725895416036026937934, 11.16978184615749421605430013517, 11.71764055761071278198965490687, 13.11640569498098506364431003441

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