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Properties

Label 2-1350-15.8-c1-0-15
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 + (0.707 − 0.707i)8-s − 5.19i·11-s + (3.67 + 3.67i)13-s − 1.00·16-s + (2.12 + 2.12i)17-s − 2i·19-s + (−3.67 + 3.67i)22-s + (−2.12 + 2.12i)23-s − 5.19i·26-s + 5.19·29-s − 5·31-s + (0.707 + 0.707i)32-s − 3i·34-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (0.250 − 0.250i)8-s − 1.56i·11-s + (1.01 + 1.01i)13-s − 0.250·16-s + (0.514 + 0.514i)17-s − 0.458i·19-s + (−0.783 + 0.783i)22-s + (−0.442 + 0.442i)23-s − 1.01i·26-s + 0.964·29-s − 0.898·31-s + (0.125 + 0.125i)32-s − 0.514i·34-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

\(L(1)\) \(\approx\) \(1.300654493\)
\(L(\frac12)\) \(\approx\) \(1.300654493\)
\(L(\frac{3}{2})\) 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 \( 1 - 7iT^{2} \)
11 \( 1 + 5.19iT - 11T^{2} \)
13 \( 1 + (-3.67 - 3.67i)T + 13iT^{2} \)
17 \( 1 + (-2.12 - 2.12i)T + 17iT^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 + (2.12 - 2.12i)T - 23iT^{2} \)
29 \( 1 - 5.19T + 29T^{2} \)
31 \( 1 + 5T + 31T^{2} \)
37 \( 1 - 37iT^{2} \)
41 \( 1 + 10.3iT - 41T^{2} \)
43 \( 1 + (-3.67 - 3.67i)T + 43iT^{2} \)
47 \( 1 + (6.36 + 6.36i)T + 47iT^{2} \)
53 \( 1 + (-8.48 + 8.48i)T - 53iT^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + (-7.34 + 7.34i)T - 67iT^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 + (-7.34 - 7.34i)T + 73iT^{2} \)
79 \( 1 + iT - 79T^{2} \)
83 \( 1 + (8.48 - 8.48i)T - 83iT^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 + (-7.34 + 7.34i)T - 97iT^{2} \)
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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

−9.416208795566714484909068233074, −8.613899829000030291033970636059, −8.254729232470676668908791881125, −7.06677006518921434841161648558, −6.22734789018604084770178650830, −5.36948322548176923705058423804, −3.94802401076731583449641794564, −3.39520106862618747619567531958, −2.05442375292574260023157002983, −0.805511050533989392355855800935, 1.08181168992992866993607110932, 2.40685977297862836793042327186, 3.75203418997097855914935939764, 4.83197753873870524577479451092, 5.67248749793375950376448710440, 6.56939567827481688559783287650, 7.39020613330307008028013195257, 8.083845422284609474055674011789, 8.814862194462685114982687176374, 9.925659013083980272968391264537

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