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Properties

Label 2-675-45.29-c2-0-22
Degree $2$
Conductor $675$
Sign $-0.287 + 0.957i$
Analytic cond. $18.3924$
Root an. cond. $4.28863$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.54 − 2.67i)2-s + (−2.77 − 4.81i)4-s + (1.91 + 1.10i)7-s − 4.80·8-s + (15.4 + 8.91i)11-s + (2.17 − 1.25i)13-s + (5.90 − 3.40i)14-s + (3.67 − 6.37i)16-s + 32.6·17-s − 7.93·19-s + (47.7 − 27.5i)22-s + (−10.6 − 18.4i)23-s − 7.77i·26-s − 12.2i·28-s + (−30.7 − 17.7i)29-s + ⋯
L(s)  = 1  + (0.772 − 1.33i)2-s + (−0.694 − 1.20i)4-s + (0.272 + 0.157i)7-s − 0.601·8-s + (1.40 + 0.810i)11-s + (0.167 − 0.0966i)13-s + (0.421 − 0.243i)14-s + (0.229 − 0.398i)16-s + 1.91·17-s − 0.417·19-s + (2.17 − 1.25i)22-s + (−0.462 − 0.801i)23-s − 0.298i·26-s − 0.437i·28-s + (−1.06 − 0.612i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.287 + 0.957i$
Analytic conductor: \(18.3924\)
Root analytic conductor: \(4.28863\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (224, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1),\ -0.287 + 0.957i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.214625691\)
\(L(\frac12)\) \(\approx\) \(3.214625691\)
\(L(2)\) 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 \( 1 + (-1.54 + 2.67i)T + (-2 - 3.46i)T^{2} \)
7 \( 1 + (-1.91 - 1.10i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-15.4 - 8.91i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-2.17 + 1.25i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 32.6T + 289T^{2} \)
19 \( 1 + 7.93T + 361T^{2} \)
23 \( 1 + (10.6 + 18.4i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (30.7 + 17.7i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (1.01 + 1.75i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 50.6iT - 1.36e3T^{2} \)
41 \( 1 + (4.65 - 2.68i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-13.4 - 7.76i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (0.943 - 1.63i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 62.0T + 2.80e3T^{2} \)
59 \( 1 + (-31.3 + 18.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-21.1 + 36.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (12.7 - 7.38i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 105. iT - 5.04e3T^{2} \)
73 \( 1 - 66.9iT - 5.32e3T^{2} \)
79 \( 1 + (34.5 - 59.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-10.6 + 18.3i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 7.16iT - 7.92e3T^{2} \)
97 \( 1 + (96.4 + 55.6i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.03704555291855026991266749518, −9.718140300903551357098433601541, −8.463831641702354461649517654547, −7.39950792215216941778941720816, −6.19687651214854646555327706856, −5.15086490691206773889625873771, −4.17476917259215376709218996878, −3.43985256332852067013167772424, −2.13026378002666776706444904776, −1.14323093087639725798917385909, 1.35208830116606419959334409885, 3.51677845140133916312166435876, 4.10234444706094671431834323284, 5.50159859719279257379471046926, 5.88391558234640037725557708812, 6.97908386462480249240205490094, 7.65881826448860423589084740164, 8.549276224118379817286200374123, 9.420592232574440342934213533367, 10.59032550451021043284814939924

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