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Properties

Label 2-675-135.49-c1-0-7
Degree $2$
Conductor $675$
Sign $0.117 + 0.993i$
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  + (−1.62 − 1.93i)2-s + (−1.32 − 1.11i)3-s + (−0.766 + 4.34i)4-s + 4.38i·6-s + (−3.01 + 0.532i)7-s + (5.28 − 3.05i)8-s + (0.520 + 2.95i)9-s + (−5.29 − 1.92i)11-s + (5.85 − 4.91i)12-s + (−2.71 + 3.23i)13-s + (5.94 + 4.98i)14-s + (−6.23 − 2.27i)16-s + (−1.43 − 0.826i)17-s + (4.88 − 5.81i)18-s + (0.120 + 0.208i)19-s + ⋯
L(s)  = 1  + (−1.15 − 1.37i)2-s + (−0.766 − 0.642i)3-s + (−0.383 + 2.17i)4-s + 1.79i·6-s + (−1.14 + 0.201i)7-s + (1.86 − 1.07i)8-s + (0.173 + 0.984i)9-s + (−1.59 − 0.581i)11-s + (1.68 − 1.41i)12-s + (−0.753 + 0.898i)13-s + (1.58 + 1.33i)14-s + (−1.55 − 0.567i)16-s + (−0.347 − 0.200i)17-s + (1.15 − 1.37i)18-s + (0.0276 + 0.0479i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.117 + 0.993i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.117 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.229894 - 0.204223i\)
\(L(\frac12)\) \(\approx\) \(0.229894 - 0.204223i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.32 + 1.11i)T \)
5 \( 1 \)
good2 \( 1 + (1.62 + 1.93i)T + (-0.347 + 1.96i)T^{2} \)
7 \( 1 + (3.01 - 0.532i)T + (6.57 - 2.39i)T^{2} \)
11 \( 1 + (5.29 + 1.92i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (2.71 - 3.23i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (1.43 + 0.826i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.120 - 0.208i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.34 - 1.29i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (-5.90 + 4.95i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-0.858 + 4.86i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (2.15 + 1.24i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.109 + 0.0918i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (0.256 - 0.705i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (-4.58 + 0.807i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 - 12.1iT - 53T^{2} \)
59 \( 1 + (4.45 - 1.62i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-2.41 - 13.6i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-4.73 + 5.64i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (-2.45 + 4.24i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.196 - 0.113i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.53 + 6.32i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (5.69 + 6.78i)T + (-14.4 + 81.7i)T^{2} \)
89 \( 1 + (-3.33 - 5.76i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.26 + 8.95i)T + (-74.3 - 62.3i)T^{2} \)
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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

−10.42879375843901552245798692224, −9.641474425286517455523392659099, −8.833972687829516890439232318991, −7.80039204682444795317381775109, −7.07356510508538193149882016141, −5.91622434183180780572628840674, −4.62004369653169846911691542393, −2.97443540934827751570052927838, −2.32169170507917870786312608845, −0.61989409660249548682358214669, 0.49223401117379135420759553065, 3.05328623206781612081605159449, 4.99160838798132816628786242078, 5.29598526685232637178216815050, 6.60233447607412222973853100112, 6.98221638394586577157883481173, 8.052742123561468298035017568349, 8.998094026680213380479084939593, 9.902857940611790602294225956561, 10.27723032272631887850413285168

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