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Properties

Label 2-675-135.49-c1-0-10
Degree $2$
Conductor $675$
Sign $0.922 - 0.385i$
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  + (−1.36 − 1.62i)2-s + (1.42 + 0.986i)3-s + (−0.430 + 2.44i)4-s + (−0.338 − 3.65i)6-s + (−0.957 + 0.168i)7-s + (0.880 − 0.508i)8-s + (1.05 + 2.80i)9-s + (0.297 + 0.108i)11-s + (−3.02 + 3.05i)12-s + (−0.973 + 1.15i)13-s + (1.57 + 1.32i)14-s + (2.63 + 0.960i)16-s + (−1.01 − 0.587i)17-s + (3.11 − 5.53i)18-s + (3.11 + 5.38i)19-s + ⋯
L(s)  = 1  + (−0.962 − 1.14i)2-s + (0.822 + 0.569i)3-s + (−0.215 + 1.22i)4-s + (−0.138 − 1.49i)6-s + (−0.361 + 0.0638i)7-s + (0.311 − 0.179i)8-s + (0.351 + 0.936i)9-s + (0.0897 + 0.0326i)11-s + (−0.872 + 0.881i)12-s + (−0.269 + 0.321i)13-s + (0.421 + 0.353i)14-s + (0.659 + 0.240i)16-s + (−0.246 − 0.142i)17-s + (0.734 − 1.30i)18-s + (0.713 + 1.23i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.922 - 0.385i$
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: \((2,\ 675,\ (\ :1/2),\ 0.922 - 0.385i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.961515 + 0.192572i\)
\(L(\frac12)\) \(\approx\) \(0.961515 + 0.192572i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.42 - 0.986i)T \)
5 \( 1 \)
good2 \( 1 + (1.36 + 1.62i)T + (-0.347 + 1.96i)T^{2} \)
7 \( 1 + (0.957 - 0.168i)T + (6.57 - 2.39i)T^{2} \)
11 \( 1 + (-0.297 - 0.108i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (0.973 - 1.15i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (1.01 + 0.587i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.11 - 5.38i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.12 + 0.375i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (-3.37 + 2.83i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (1.50 - 8.54i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (-3.86 - 2.23i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.47 - 3.75i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (1.91 - 5.25i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (-2.43 + 0.429i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 - 10.8iT - 53T^{2} \)
59 \( 1 + (1.62 - 0.589i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-0.176 - 0.999i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (0.550 - 0.656i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (-4.79 + 8.31i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-13.1 + 7.62i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.59 + 7.20i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (3.01 + 3.58i)T + (-14.4 + 81.7i)T^{2} \)
89 \( 1 + (7.74 + 13.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.89 - 5.21i)T + (-74.3 - 62.3i)T^{2} \)
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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

−10.31708491018943916109177039981, −9.712666503285836927096266480157, −9.157511566142836186728828830416, −8.270718374793651175648176096242, −7.57441623807657332271583901703, −6.11937406048022772851364833586, −4.68120593824683299595332654929, −3.52168726975458874802697902556, −2.72396868553313850815985098257, −1.53903307418447762180925539612, 0.67633313197859117273193058711, 2.48487937370958726783925272005, 3.75910990145637539849630947830, 5.37094241963712067286976807553, 6.47856950718521362413345320077, 7.06573786109093682185296374337, 7.85593856364442504154430789898, 8.526697655913526913874440222563, 9.439526259870678926828202416773, 9.791445873977435616009374421723

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