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Properties

Label 2-1350-15.8-c1-0-22
Degree $2$
Conductor $1350$
Sign $0.525 + 0.850i$
Analytic cond. $10.7798$
Root an. cond. $3.28326$
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 + 1.00i·4-s + (0.366 − 0.366i)7-s + (−0.707 + 0.707i)8-s − 4.76i·11-s + (−3.46 − 3.46i)13-s + 0.517·14-s − 1.00·16-s + (−1.03 − 1.03i)17-s − 7.46i·19-s + (3.36 − 3.36i)22-s + (−1.41 + 1.41i)23-s − 4.89i·26-s + (0.366 + 0.366i)28-s − 6.31·29-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (0.138 − 0.138i)7-s + (−0.250 + 0.250i)8-s − 1.43i·11-s + (−0.960 − 0.960i)13-s + 0.138·14-s − 0.250·16-s + (−0.251 − 0.251i)17-s − 1.71i·19-s + (0.717 − 0.717i)22-s + (−0.294 + 0.294i)23-s − 0.960i·26-s + (0.0691 + 0.0691i)28-s − 1.17·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.525 + 0.850i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ 0.525 + 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.587046278\)
\(L(\frac12)\) \(\approx\) \(1.587046278\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-0.366 + 0.366i)T - 7iT^{2} \)
11 \( 1 + 4.76iT - 11T^{2} \)
13 \( 1 + (3.46 + 3.46i)T + 13iT^{2} \)
17 \( 1 + (1.03 + 1.03i)T + 17iT^{2} \)
19 \( 1 + 7.46iT - 19T^{2} \)
23 \( 1 + (1.41 - 1.41i)T - 23iT^{2} \)
29 \( 1 + 6.31T + 29T^{2} \)
31 \( 1 - 6.66T + 31T^{2} \)
37 \( 1 + (-2.19 + 2.19i)T - 37iT^{2} \)
41 \( 1 - 4.62iT - 41T^{2} \)
43 \( 1 + (-1.26 - 1.26i)T + 43iT^{2} \)
47 \( 1 + (4.24 + 4.24i)T + 47iT^{2} \)
53 \( 1 + (-5.08 + 5.08i)T - 53iT^{2} \)
59 \( 1 + 4.52T + 59T^{2} \)
61 \( 1 - 9.46T + 61T^{2} \)
67 \( 1 + (-6.19 + 6.19i)T - 67iT^{2} \)
71 \( 1 - 15.4iT - 71T^{2} \)
73 \( 1 + (9.36 + 9.36i)T + 73iT^{2} \)
79 \( 1 + 3.46iT - 79T^{2} \)
83 \( 1 + (-1.46 + 1.46i)T - 83iT^{2} \)
89 \( 1 + 5.93T + 89T^{2} \)
97 \( 1 + (-13.5 + 13.5i)T - 97iT^{2} \)
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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

−9.340885126872597045541195133711, −8.522194393168299046623009101555, −7.77422080409304042857905085265, −7.01661761284084757514628213386, −6.08218228645337886172655567981, −5.30426531764726958799187353079, −4.53173643196275149924772110189, −3.33327990843366005362523510335, −2.55321983511391091952032180108, −0.52989045453619554196041620153, 1.70440623496560762538131849006, 2.39859121767611385391105765859, 3.86526893820192050926517767123, 4.49570910410841189890190032571, 5.38750303816109615467094061474, 6.39917616749354757458899226766, 7.22401906554408921200183250932, 8.066972305737797121484982838998, 9.188100111294749919393969683523, 9.909497203266547165135419229266

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