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Properties

Label 2-675-135.49-c1-0-28
Degree $2$
Conductor $675$
Sign $-0.865 - 0.501i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 - 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.865 - 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.865 - 0.501i$
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: \((2,\ 675,\ (\ :1/2),\ -0.865 - 0.501i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.912703 + 3.39421i\)
\(L(\frac12)\) \(\approx\) \(0.912703 + 3.39421i\)
\(L(\frac{3}{2})\) 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

−10.77271000708974388632975243310, −10.04678566461786229905750285130, −8.376615474006188502557979786107, −8.112207589054028994927766709683, −7.66779727235471810849593044651, −6.10045618980844807488081763959, −5.36349336157572243850622552079, −4.58880572455867920206105068126, −3.69112664571016319352659580784, −2.59296822880133799517056746840, 1.48445222708352362642519633307, 2.24135412528041975636787624262, 3.25350824375078114235048697771, 4.41627608687596871065824102618, 5.17927980558880865999322663180, 6.31470314610668691531154384607, 7.65981712342045639784283407560, 8.409173607697516748365266946613, 9.537835208141581391551842581101, 10.42725738677588412562146941051

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