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Properties

Label 2-675-45.2-c1-0-4
Degree $2$
Conductor $675$
Sign $-0.0855 - 0.996i$
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  + (0.347 + 1.29i)2-s + (0.170 − 0.0981i)4-s + (−1.97 + 0.530i)7-s + (2.08 + 2.08i)8-s + (0.762 + 0.440i)11-s + (5.36 + 1.43i)13-s + (−1.37 − 2.38i)14-s + (−1.78 + 3.09i)16-s + (−1.13 + 1.13i)17-s + 1.52i·19-s + (−0.305 + 1.14i)22-s + (0.410 − 1.53i)23-s + 7.46i·26-s + (−0.284 + 0.284i)28-s + (−0.796 + 1.37i)29-s + ⋯
L(s)  = 1  + (0.245 + 0.917i)2-s + (0.0850 − 0.0490i)4-s + (−0.747 + 0.200i)7-s + (0.737 + 0.737i)8-s + (0.229 + 0.132i)11-s + (1.48 + 0.398i)13-s + (−0.367 − 0.636i)14-s + (−0.446 + 0.772i)16-s + (−0.275 + 0.275i)17-s + 0.349i·19-s + (−0.0652 + 0.243i)22-s + (0.0856 − 0.319i)23-s + 1.46i·26-s + (−0.0537 + 0.0537i)28-s + (−0.147 + 0.256i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0855 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0855 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.0855 - 0.996i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (332, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ -0.0855 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28117 + 1.39588i\)
\(L(\frac12)\) \(\approx\) \(1.28117 + 1.39588i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-0.347 - 1.29i)T + (-1.73 + i)T^{2} \)
7 \( 1 + (1.97 - 0.530i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.762 - 0.440i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-5.36 - 1.43i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (1.13 - 1.13i)T - 17iT^{2} \)
19 \( 1 - 1.52iT - 19T^{2} \)
23 \( 1 + (-0.410 + 1.53i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (0.796 - 1.37i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.49 - 6.05i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.25 - 4.25i)T + 37iT^{2} \)
41 \( 1 + (3.11 - 1.79i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.497 + 1.85i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (2.14 + 7.99i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-4.65 - 4.65i)T + 53iT^{2} \)
59 \( 1 + (3.81 + 6.61i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.64 + 11.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.859 + 3.20i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 5.89iT - 71T^{2} \)
73 \( 1 + (1.58 - 1.58i)T - 73iT^{2} \)
79 \( 1 + (6.69 + 3.86i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.59 - 2.57i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 - 4.62T + 89T^{2} \)
97 \( 1 + (3.82 - 1.02i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.74671508579251164433116994296, −9.837669859059221167888017848312, −8.729676202580942765858215461684, −8.119512718521149604683385785806, −6.80357748792356143850272848826, −6.45587512375428008446467429246, −5.57676714411665073475623136112, −4.42865085740645380447591295508, −3.24293490464323475776644402398, −1.65312057744106074350349890889, 1.03899119220757347241745641545, 2.55901305366765752677062187822, 3.52492859111147167831511156084, 4.29097370097850902943776469429, 5.83193228249979504983179921382, 6.65740564975608045698149920739, 7.61500516473920594435191229459, 8.701732002970083337934700754402, 9.675336079280862991904299972105, 10.40301894873101090850644553227

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