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Properties

Label 2-675-9.4-c1-0-12
Degree $2$
Conductor $675$
Sign $-0.144 + 0.989i$
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  + (−0.236 − 0.409i)2-s + (0.888 − 1.53i)4-s + (1.28 + 2.21i)7-s − 1.78·8-s + (−3.08 − 5.34i)11-s + (1.06 − 1.84i)13-s + (0.606 − 1.05i)14-s + (−1.35 − 2.34i)16-s + 3.16·17-s + 0.356·19-s + (−1.45 + 2.52i)22-s + (2.10 − 3.64i)23-s − 1.00·26-s + 4.55·28-s + (0.843 + 1.46i)29-s + ⋯
L(s)  = 1  + (−0.167 − 0.289i)2-s + (0.444 − 0.769i)4-s + (0.484 + 0.838i)7-s − 0.631·8-s + (−0.929 − 1.61i)11-s + (0.295 − 0.512i)13-s + (0.162 − 0.280i)14-s + (−0.338 − 0.585i)16-s + 0.768·17-s + 0.0817·19-s + (−0.311 + 0.539i)22-s + (0.439 − 0.760i)23-s − 0.197·26-s + 0.860·28-s + (0.156 + 0.271i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.144 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.144 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.144 + 0.989i$
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: \((2,\ 675,\ (\ :1/2),\ -0.144 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.928426 - 1.07438i\)
\(L(\frac12)\) \(\approx\) \(0.928426 - 1.07438i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.236 + 0.409i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-1.28 - 2.21i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.08 + 5.34i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.06 + 1.84i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.16T + 17T^{2} \)
19 \( 1 - 0.356T + 19T^{2} \)
23 \( 1 + (-2.10 + 3.64i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.843 - 1.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.12 + 7.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.63T + 37T^{2} \)
41 \( 1 + (1.36 - 2.36i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.83 + 6.64i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.71 - 9.89i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 9.43T + 53T^{2} \)
59 \( 1 + (-5.10 + 8.84i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.00549 - 0.00952i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.491 + 0.851i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.43T + 71T^{2} \)
73 \( 1 + 6.61T + 73T^{2} \)
79 \( 1 + (-4.73 - 8.20i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.20 - 9.02i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6.26T + 89T^{2} \)
97 \( 1 + (-3.60 - 6.24i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.47587652853216474722301467622, −9.475761829480163921314444309943, −8.506947314661654855746862964472, −7.917025037883148633463159904411, −6.46217038934009036195681256969, −5.68052699209653494082797287532, −5.10703072714107634025395719975, −3.26352676261105135842224032789, −2.38136835969692106340247236280, −0.813393206786288813169961555785, 1.74609353651287406568692646933, 3.10613465705889698943306200679, 4.26726916748150760318806178598, 5.21104963207659769815476471677, 6.64803455321212821817537779799, 7.36052922256035611320872673163, 7.85561739220693587483577072140, 8.862168342551111977582267531956, 9.980390991171374422936629774966, 10.62461606835487055602585372337

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