Properties

Label 2-675-45.4-c1-0-14
Degree $2$
Conductor $675$
Sign $-0.173 + 0.984i$
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  + (2.17 − 1.25i)2-s + (2.16 − 3.74i)4-s + (0.445 − 0.257i)7-s − 5.83i·8-s + (−1.66 − 2.87i)11-s + (1.14 + 0.660i)13-s + (0.646 − 1.11i)14-s + (−3.01 − 5.22i)16-s − 3.32i·17-s + 1.32·19-s + (−7.23 − 4.17i)22-s + (3.57 + 2.06i)23-s + 3.32·26-s − 2.22i·28-s + (0.693 + 1.20i)29-s + ⋯
L(s)  = 1  + (1.53 − 0.888i)2-s + (1.08 − 1.87i)4-s + (0.168 − 0.0971i)7-s − 2.06i·8-s + (−0.500 − 0.867i)11-s + (0.317 + 0.183i)13-s + (0.172 − 0.299i)14-s + (−0.753 − 1.30i)16-s − 0.805i·17-s + 0.303·19-s + (−1.54 − 0.890i)22-s + (0.745 + 0.430i)23-s + 0.651·26-s − 0.419i·28-s + (0.128 + 0.222i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.173 + 0.984i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ -0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.24720 - 2.67658i\)
\(L(\frac12)\) \(\approx\) \(2.24720 - 2.67658i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.17 + 1.25i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (-0.445 + 0.257i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.66 + 2.87i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.14 - 0.660i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.32iT - 17T^{2} \)
19 \( 1 - 1.32T + 19T^{2} \)
23 \( 1 + (-3.57 - 2.06i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.693 - 1.20i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.36 - 7.56i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.292iT - 37T^{2} \)
41 \( 1 + (5.67 - 9.82i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-8.96 + 5.17i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.21 - 2.43i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 5.02iT - 53T^{2} \)
59 \( 1 + (-2.51 + 4.35i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.67 + 6.36i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.18 - 4.72i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.99T + 71T^{2} \)
73 \( 1 - 6.05iT - 73T^{2} \)
79 \( 1 + (4.02 + 6.97i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.33 - 0.771i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (-10.6 + 6.12i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.77636746016614252115957852643, −9.717478785567000504109780067870, −8.652650255493903530224016143831, −7.37031075551879719807328967850, −6.31660437413310982494440554481, −5.38008820942690119810034767368, −4.72630648407222491042001440531, −3.49418858705404238656705331095, −2.81029030747697226434653694714, −1.32895331443057879378034178288, 2.25462046693379687792684979336, 3.51951694084309030185261747079, 4.44264299807973011108202036468, 5.30933779711155332134766974443, 6.08299669691825453195240816361, 7.06585483314038871581833460375, 7.73540956216490526615462973285, 8.675137931041084939755636837470, 9.963437159170939042905692418716, 11.00836759845950033679949786291

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