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Properties

Label 2-675-27.22-c1-0-7
Degree $2$
Conductor $675$
Sign $-0.948 - 0.317i$
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  + (−1.84 + 1.55i)2-s + (1.53 + 0.794i)3-s + (0.665 − 3.77i)4-s + (−4.07 + 0.920i)6-s + (0.0583 + 0.330i)7-s + (2.20 + 3.82i)8-s + (1.73 + 2.44i)9-s + (−4.97 − 1.81i)11-s + (4.01 − 5.27i)12-s + (4.68 + 3.93i)13-s + (−0.621 − 0.521i)14-s + (−2.82 − 1.02i)16-s + (−1.52 + 2.63i)17-s + (−7.01 − 1.82i)18-s + (−0.260 − 0.450i)19-s + ⋯
L(s)  = 1  + (−1.30 + 1.09i)2-s + (0.888 + 0.458i)3-s + (0.332 − 1.88i)4-s + (−1.66 + 0.375i)6-s + (0.0220 + 0.125i)7-s + (0.780 + 1.35i)8-s + (0.579 + 0.814i)9-s + (−1.49 − 0.545i)11-s + (1.16 − 1.52i)12-s + (1.29 + 1.09i)13-s + (−0.166 − 0.139i)14-s + (−0.706 − 0.257i)16-s + (−0.368 + 0.638i)17-s + (−1.65 − 0.429i)18-s + (−0.0597 − 0.103i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.317i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 - 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.948 - 0.317i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ -0.948 - 0.317i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.139164 + 0.853176i\)
\(L(\frac12)\) \(\approx\) \(0.139164 + 0.853176i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.53 - 0.794i)T \)
5 \( 1 \)
good2 \( 1 + (1.84 - 1.55i)T + (0.347 - 1.96i)T^{2} \)
7 \( 1 + (-0.0583 - 0.330i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (4.97 + 1.81i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (-4.68 - 3.93i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (1.52 - 2.63i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.260 + 0.450i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.0350 - 0.198i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-1.41 + 1.18i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (1.58 - 8.98i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (3.11 - 5.39i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.96 - 5.84i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (6.38 + 2.32i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-2.11 - 12.0i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 - 2.74T + 53T^{2} \)
59 \( 1 + (6.56 - 2.38i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (2.16 + 12.2i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (4.29 + 3.60i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-1.58 + 2.74i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.731 - 1.26i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.18 - 1.83i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (0.578 - 0.485i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (3.18 + 5.50i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.0 + 3.64i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.60933793944456939726355407132, −9.730141531995622623447402246353, −8.797164786093510767483140788615, −8.468905928219956551087251835405, −7.71635328858852406557908514966, −6.71713022162278874730062000481, −5.80629448787207396992071523114, −4.62722712410182522861614692333, −3.18117375301675183842217422090, −1.63609363472378331415879940455, 0.64409053044849407600159958944, 2.10149981282098793820612324571, 2.87422777887060052949276200218, 3.92325308700155062333517369886, 5.65241364020071873950786387431, 7.24030105396188949296801310490, 7.79927324365801666524053348024, 8.532514777441682370589947825041, 9.202720271134829870285345652158, 10.21984526915896140886591445704

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