Properties

Label 2-675-9.4-c1-0-6
Degree $2$
Conductor $675$
Sign $-0.731 - 0.681i$
Analytic cond. $5.38990$
Root an. cond. $2.32161$
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  + (1.04 + 1.80i)2-s + (−1.17 + 2.03i)4-s + (2.04 + 3.53i)7-s − 0.734·8-s + (−0.675 − 1.17i)11-s + (0.324 − 0.561i)13-s + (−4.26 + 7.38i)14-s + (1.58 + 2.74i)16-s − 1.35·17-s + 0.648·19-s + (1.41 − 2.44i)22-s + (−2.39 + 4.14i)23-s + 1.35·26-s − 9.61·28-s + (1.93 + 3.35i)29-s + ⋯
L(s)  = 1  + (0.737 + 1.27i)2-s + (−0.587 + 1.01i)4-s + (0.772 + 1.33i)7-s − 0.259·8-s + (−0.203 − 0.353i)11-s + (0.0898 − 0.155i)13-s + (−1.13 + 1.97i)14-s + (0.396 + 0.686i)16-s − 0.327·17-s + 0.148·19-s + (0.300 − 0.520i)22-s + (−0.499 + 0.864i)23-s + 0.265·26-s − 1.81·28-s + (0.359 + 0.623i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.731 - 0.681i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.731 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.731 - 0.681i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ -0.731 - 0.681i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.857724 + 2.17905i\)
\(L(\frac12)\) \(\approx\) \(0.857724 + 2.17905i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-1.04 - 1.80i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-2.04 - 3.53i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.675 + 1.17i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.324 + 0.561i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.35T + 17T^{2} \)
19 \( 1 - 0.648T + 19T^{2} \)
23 \( 1 + (2.39 - 4.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.93 - 3.35i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.84 + 6.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.52T + 37T^{2} \)
41 \( 1 + (0.0898 - 0.155i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.410 + 0.710i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.45 + 9.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 4.17T + 53T^{2} \)
59 \( 1 + (-2.08 + 3.61i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.91 - 3.30i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.07 + 7.05i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.11T + 71T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 + (5.17 + 8.95i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.12 - 10.6i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + (6.79 + 11.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.00073628076886972288213829194, −9.804612347819256643252460096700, −8.595840366500223327182071590753, −8.210865150534318111698634496630, −7.19255248551838519091052026396, −6.18430758467416288617659408359, −5.45527944220360320357748034937, −4.85272215278122223556681619845, −3.55530562150653588454359337568, −2.03114645312125324038751132377, 1.09400445785701086496586887870, 2.28573164272010022827509085849, 3.58336647593880266171882191332, 4.44044203061591774518737120080, 5.04417334236876156555514805270, 6.59315371632598908634299970123, 7.55649236473732089647968734036, 8.454194027182124167696840263415, 9.839068588663890039645758185879, 10.44310386278615968036993698303

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