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Properties

Label 2-1350-15.2-c1-0-20
Degree $2$
Conductor $1350$
Sign $-0.793 + 0.608i$
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 + (−1.22 − 1.22i)7-s + (−0.707 − 0.707i)8-s − 1.73·14-s − 1.00·16-s + (4.24 − 4.24i)17-s − i·19-s + (−4.24 − 4.24i)23-s + (−1.22 + 1.22i)28-s − 10.3·29-s + 7·31-s + (−0.707 + 0.707i)32-s − 6i·34-s + (−6.12 − 6.12i)37-s + (−0.707 − 0.707i)38-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.462 − 0.462i)7-s + (−0.250 − 0.250i)8-s − 0.462·14-s − 0.250·16-s + (1.02 − 1.02i)17-s − 0.229i·19-s + (−0.884 − 0.884i)23-s + (−0.231 + 0.231i)28-s − 1.92·29-s + 1.25·31-s + (−0.125 + 0.125i)32-s − 1.02i·34-s + (−1.00 − 1.00i)37-s + (−0.114 − 0.114i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.793 + 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.793 + 0.608i$
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: \((2,\ 1350,\ (\ :1/2),\ -0.793 + 0.608i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.544401617\)
\(L(\frac12)\) \(\approx\) \(1.544401617\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.22 + 1.22i)T + 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + (-4.24 + 4.24i)T - 17iT^{2} \)
19 \( 1 + iT - 19T^{2} \)
23 \( 1 + (4.24 + 4.24i)T + 23iT^{2} \)
29 \( 1 + 10.3T + 29T^{2} \)
31 \( 1 - 7T + 31T^{2} \)
37 \( 1 + (6.12 + 6.12i)T + 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (-1.22 + 1.22i)T - 43iT^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + (8.48 + 8.48i)T + 53iT^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 - 5T + 61T^{2} \)
67 \( 1 + (-7.34 - 7.34i)T + 67iT^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + (-8.57 + 8.57i)T - 73iT^{2} \)
79 \( 1 - 13iT - 79T^{2} \)
83 \( 1 + (4.24 + 4.24i)T + 83iT^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 + (6.12 + 6.12i)T + 97iT^{2} \)
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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

−9.640781582592792370779018186985, −8.557397630844373902164986656819, −7.54231711279600932802244772172, −6.78504910854424318411910996414, −5.82168114760504110087184075426, −5.00440745377781370645426742002, −3.97105957240062381972839336065, −3.19627476439032348936910898264, −2.05420660940367621869200611317, −0.52086423632855374155832590026, 1.74389751735763430914934563273, 3.15164163535647437438644188656, 3.85623985475172646139031386559, 5.02293640430466324230746145659, 5.91190136839700195583402435546, 6.36617705079859062342576601375, 7.61105920595576213222938491717, 8.044764064581457021738410798613, 9.113575167388213321734637591073, 9.800299414783516801438315554807

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