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Properties

Label 2-675-9.7-c1-0-3
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: $\chi_{675} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ -0.144 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.892528 + 1.03284i\)
\(L(\frac12)\) \(\approx\) \(0.892528 + 1.03284i\)
\(L(\frac{3}{2})\) 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

−10.65328001459690587000045441576, −10.06258266979346672839183621413, −8.983652248131037101074723723517, −8.067716023885084528210221916637, −7.25888309922819597290120005543, −6.41830226091935261091237172853, −5.12969232948471379106441292383, −4.22236541335488726787408480973, −2.78823388648393718032501100768, −2.22800276076251992325285286081, 0.64744647800304890071788867040, 2.37083798407704941980822581505, 3.68891547011130357371936929907, 4.89863801434523948215604053937, 5.89044897013248218034741783006, 6.58105052933723238282162161621, 7.50596824528879870941994969265, 8.406247124034758290526620517760, 9.639326251180183186306725896549, 10.24630517975625281672686779321

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