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Properties

Label 2-675-27.16-c1-0-32
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: $\chi_{675} (151, \cdot )$
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

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

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