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Properties

Label 2-675-9.4-c1-0-3
Degree $2$
Conductor $675$
Sign $-0.951 + 0.307i$
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  + (1.31 + 2.28i)2-s + (−2.46 + 4.27i)4-s + (0.898 + 1.55i)7-s − 7.73·8-s + (0.904 + 1.56i)11-s + (−0.985 + 1.70i)13-s + (−2.36 + 4.09i)14-s + (−5.24 − 9.08i)16-s − 4.80·17-s + 2.96·19-s + (−2.38 + 4.12i)22-s + (−0.866 + 1.50i)23-s − 5.19·26-s − 8.87·28-s + (−3.68 − 6.38i)29-s + ⋯
L(s)  = 1  + (0.931 + 1.61i)2-s + (−1.23 + 2.13i)4-s + (0.339 + 0.588i)7-s − 2.73·8-s + (0.272 + 0.472i)11-s + (−0.273 + 0.473i)13-s + (−0.632 + 1.09i)14-s + (−1.31 − 2.27i)16-s − 1.16·17-s + 0.680·19-s + (−0.507 + 0.879i)22-s + (−0.180 + 0.313i)23-s − 1.01·26-s − 1.67·28-s + (−0.684 − 1.18i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.307i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 + 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.951 + 0.307i$
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: \((2,\ 675,\ (\ :1/2),\ -0.951 + 0.307i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.318109 - 2.01884i\)
\(L(\frac12)\) \(\approx\) \(0.318109 - 2.01884i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.31 - 2.28i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-0.898 - 1.55i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.904 - 1.56i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.985 - 1.70i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.80T + 17T^{2} \)
19 \( 1 - 2.96T + 19T^{2} \)
23 \( 1 + (0.866 - 1.50i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.68 + 6.38i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.31 + 2.27i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 11.6T + 37T^{2} \)
41 \( 1 + (1.23 - 2.13i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.63 - 6.30i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.14 - 5.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 1.72T + 53T^{2} \)
59 \( 1 + (-5.51 + 9.54i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.33 - 10.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.55 - 7.88i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.27T + 71T^{2} \)
73 \( 1 + 3.58T + 73T^{2} \)
79 \( 1 + (1.05 + 1.82i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.549 - 0.951i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 13.2T + 89T^{2} \)
97 \( 1 + (-1.91 - 3.31i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.41176374235122599873754813143, −9.668162038035737895873617107615, −9.000849186726305115247186759138, −8.015999148395461536212889504323, −7.35829793444150885383073515995, −6.40677853968808858586539701857, −5.71703535064622024450136774286, −4.67424529596192635529559634724, −4.05159814834040768740864101118, −2.50046094299090050520872735498, 0.832939376532421252641380507555, 2.19108602951918060473080479406, 3.31523592606609423583759737796, 4.22143161909657374090227309186, 5.06451448468349899660773528532, 6.03386518850729853519841483446, 7.30345525961063249034980621727, 8.673868958016268344034538616182, 9.487663720514676355965246824029, 10.44855436429833799926517807648

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