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Properties

Label 2-675-5.3-c0-0-0
Degree $2$
Conductor $675$
Sign $-0.130 - 0.991i$
Analytic cond. $0.336868$
Root an. cond. $0.580404$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·4-s + (−1.22 + 1.22i)7-s − 16-s + i·19-s + (−1.22 − 1.22i)28-s + 31-s + (1.22 − 1.22i)37-s + (1.22 + 1.22i)43-s − 1.99i·49-s − 61-s − i·64-s + (1.22 + 1.22i)73-s − 76-s − i·79-s + (1.22 − 1.22i)97-s + ⋯
L(s)  = 1  + i·4-s + (−1.22 + 1.22i)7-s − 16-s + i·19-s + (−1.22 − 1.22i)28-s + 31-s + (1.22 − 1.22i)37-s + (1.22 + 1.22i)43-s − 1.99i·49-s − 61-s − i·64-s + (1.22 + 1.22i)73-s − 76-s − i·79-s + (1.22 − 1.22i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.130 - 0.991i$
Analytic conductor: \(0.336868\)
Root analytic conductor: \(0.580404\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (568, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :0),\ -0.130 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7698692269\)
\(L(\frac12)\) \(\approx\) \(0.7698692269\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 - iT^{2} \)
7 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 - iT - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
79 \( 1 + iT - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
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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

−11.08036165897135702940569209781, −9.833023799444820084607818085709, −9.218851002014410834601413769992, −8.383135414095142593224134500463, −7.54596984993223564554812984324, −6.42899130769434689778920533763, −5.75821690281665609104477244565, −4.31783847928371327170890814106, −3.23925541200774915063674261491, −2.42443957780060486738654765509, 0.871844952090844348394648054402, 2.69714876681067840061640995031, 3.99858547274640625115635070824, 4.95749602967939851548985725555, 6.23335339877002256120005323054, 6.72131154433993703450403957201, 7.69048161658076330804620024888, 9.071842465931902905560325662704, 9.715117395904601483239910268589, 10.42622882242498363016959179455

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