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Properties

Label 2-675-45.29-c2-0-8
Degree $2$
Conductor $675$
Sign $0.709 - 0.704i$
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  + (0.346 − 0.600i)2-s + (1.75 + 3.04i)4-s + (−10.6 − 6.14i)7-s + 5.21·8-s + (11.5 + 6.67i)11-s + (1.49 − 0.865i)13-s + (−7.37 + 4.25i)14-s + (−5.23 + 9.06i)16-s − 5.84·17-s + 16.8·19-s + (8.00 − 4.62i)22-s + (16.6 + 28.8i)23-s − 1.19i·26-s − 43.2i·28-s + (28.4 + 16.4i)29-s + ⋯
L(s)  = 1  + (0.173 − 0.300i)2-s + (0.439 + 0.762i)4-s + (−1.51 − 0.877i)7-s + 0.651·8-s + (1.05 + 0.606i)11-s + (0.115 − 0.0665i)13-s + (−0.526 + 0.303i)14-s + (−0.327 + 0.566i)16-s − 0.343·17-s + 0.884·19-s + (0.364 − 0.210i)22-s + (0.725 + 1.25i)23-s − 0.0461i·26-s − 1.54i·28-s + (0.980 + 0.565i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.709 - 0.704i$
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: \((2,\ 675,\ (\ :1),\ 0.709 - 0.704i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.948962828\)
\(L(\frac12)\) \(\approx\) \(1.948962828\)
\(L(2)\) 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 \( 1 + (-0.346 + 0.600i)T + (-2 - 3.46i)T^{2} \)
7 \( 1 + (10.6 + 6.14i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-11.5 - 6.67i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-1.49 + 0.865i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 5.84T + 289T^{2} \)
19 \( 1 - 16.8T + 361T^{2} \)
23 \( 1 + (-16.6 - 28.8i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-28.4 - 16.4i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (0.0240 + 0.0415i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 29.7iT - 1.36e3T^{2} \)
41 \( 1 + (8.34 - 4.81i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-14.6 - 8.45i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (3.63 - 6.30i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 29.6T + 2.80e3T^{2} \)
59 \( 1 + (79.7 - 46.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-10.0 + 17.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (30.2 - 17.4i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 83.7iT - 5.04e3T^{2} \)
73 \( 1 - 7.39iT - 5.32e3T^{2} \)
79 \( 1 + (-10.6 + 18.3i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-29.6 + 51.3i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 46.7iT - 7.92e3T^{2} \)
97 \( 1 + (20.0 + 11.5i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.37106258258477372295718848383, −9.649633067767419184881609130006, −8.821084099732618198171658375577, −7.44833805023414242444750363361, −7.00050340561442086947664828362, −6.19594348285691385879296744438, −4.58023031602690654139593204774, −3.60552057567306503761244745842, −2.98191447726157424419942435233, −1.28615910870869706355850967628, 0.73337846324628253706882878443, 2.39775189169071387107581262239, 3.46266070656649800464002228834, 4.85466346044595530546190314228, 6.06200761191491495109139484217, 6.33298238794008493720024496054, 7.22152204213329795606692535576, 8.707507481059591747350206237424, 9.317529930459807977269889050703, 10.09772648457193626679561513636

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