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Properties

Label 2-1350-15.2-c1-0-6
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: $\chi_{1350} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ 0.525 - 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.192728648\)
\(L(\frac12)\) \(\approx\) \(1.192728648\)
\(L(\frac{3}{2})\) 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

−9.927941001410014754800596664297, −8.615376727266055845708097088447, −8.348003028490573609227473312586, −7.33781160921093333590721991767, −6.48511343983482037276005277859, −5.73650444583437162278894032787, −4.85220997425000641415401527888, −3.69664873482844562861093778945, −2.45918251238146373361815162972, −1.03762372211956648975650616143, 0.73340809871335329737642347724, 2.25944493970122890474708629635, 2.99032685533505186678124107982, 4.49681970818806178366183599081, 4.92192719282493834171821220260, 6.36690853786085095723521594337, 7.30668339318953142741742562958, 7.78004806138461704733426174580, 8.836421059239605392845182194078, 9.566990172825196891555533342438

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