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Properties

Label 2-675-5.4-c3-0-21
Degree $2$
Conductor $675$
Sign $-0.894 - 0.447i$
Analytic cond. $39.8262$
Root an. cond. $6.31080$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.45i·2-s − 21.7·4-s − 11.8i·7-s − 75.3i·8-s − 56.2·11-s − 34.5i·13-s + 64.4·14-s + 236.·16-s + 39.2i·17-s + 146.·19-s − 306. i·22-s + 23.5i·23-s + 188.·26-s + 257. i·28-s − 161.·29-s + ⋯
L(s)  = 1  + 1.92i·2-s − 2.72·4-s − 0.637i·7-s − 3.32i·8-s − 1.54·11-s − 0.738i·13-s + 1.23·14-s + 3.69·16-s + 0.560i·17-s + 1.76·19-s − 2.97i·22-s + 0.213i·23-s + 1.42·26-s + 1.73i·28-s − 1.03·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(39.8262\)
Root analytic conductor: \(6.31080\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :3/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.155197725\)
\(L(\frac12)\) \(\approx\) \(1.155197725\)
\(L(\frac{5}{2})\) 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 \( 1 - 5.45iT - 8T^{2} \)
7 \( 1 + 11.8iT - 343T^{2} \)
11 \( 1 + 56.2T + 1.33e3T^{2} \)
13 \( 1 + 34.5iT - 2.19e3T^{2} \)
17 \( 1 - 39.2iT - 4.91e3T^{2} \)
19 \( 1 - 146.T + 6.85e3T^{2} \)
23 \( 1 - 23.5iT - 1.21e4T^{2} \)
29 \( 1 + 161.T + 2.43e4T^{2} \)
31 \( 1 + 29.5T + 2.97e4T^{2} \)
37 \( 1 + 217. iT - 5.06e4T^{2} \)
41 \( 1 - 142.T + 6.89e4T^{2} \)
43 \( 1 - 468. iT - 7.95e4T^{2} \)
47 \( 1 - 394. iT - 1.03e5T^{2} \)
53 \( 1 - 134. iT - 1.48e5T^{2} \)
59 \( 1 - 131.T + 2.05e5T^{2} \)
61 \( 1 - 259.T + 2.26e5T^{2} \)
67 \( 1 - 445. iT - 3.00e5T^{2} \)
71 \( 1 - 560.T + 3.57e5T^{2} \)
73 \( 1 - 88.6iT - 3.89e5T^{2} \)
79 \( 1 + 450.T + 4.93e5T^{2} \)
83 \( 1 + 284. iT - 5.71e5T^{2} \)
89 \( 1 + 625.T + 7.04e5T^{2} \)
97 \( 1 + 193. iT - 9.12e5T^{2} \)
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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

−10.07889714367961767245715171875, −9.429939783554655683397932046796, −8.295849569298943883120896315530, −7.59074362674923412081783211214, −7.27334620286812696343617334427, −5.85345036896963174862420900701, −5.43415493827552756439719171151, −4.41347508091482497493025034901, −3.23937958933816394037819488616, −0.75055701109972471154351309615, 0.51549971028572369068865599237, 1.96526773079395764424878919358, 2.76722453821194036146583946864, 3.71189172593715866074471781757, 5.03268829739542667404139080346, 5.46496853803574928752092335671, 7.39392379688534756932996061360, 8.413202639660499075625611339100, 9.243626656394611539229650916087, 9.886162387374655296553218541128

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