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Properties

Label 2-675-5.4-c3-0-62
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  − 2.12i·2-s + 3.47·4-s − 30.7i·7-s − 24.4i·8-s + 50.1·11-s − 15.9i·13-s − 65.2·14-s − 24.0·16-s − 105. i·17-s + 21.3·19-s − 106. i·22-s + 136. i·23-s − 33.9·26-s − 106. i·28-s + 224.·29-s + ⋯
L(s)  = 1  − 0.751i·2-s + 0.434·4-s − 1.65i·7-s − 1.07i·8-s + 1.37·11-s − 0.340i·13-s − 1.24·14-s − 0.375·16-s − 1.50i·17-s + 0.257·19-s − 1.03i·22-s + 1.23i·23-s − 0.255·26-s − 0.720i·28-s + 1.43·29-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

\(L(2)\) \(\approx\) \(2.506937951\)
\(L(\frac12)\) \(\approx\) \(2.506937951\)
\(L(\frac{5}{2})\) 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 \( 1 + 2.12iT - 8T^{2} \)
7 \( 1 + 30.7iT - 343T^{2} \)
11 \( 1 - 50.1T + 1.33e3T^{2} \)
13 \( 1 + 15.9iT - 2.19e3T^{2} \)
17 \( 1 + 105. iT - 4.91e3T^{2} \)
19 \( 1 - 21.3T + 6.85e3T^{2} \)
23 \( 1 - 136. iT - 1.21e4T^{2} \)
29 \( 1 - 224.T + 2.43e4T^{2} \)
31 \( 1 + 225.T + 2.97e4T^{2} \)
37 \( 1 - 416. iT - 5.06e4T^{2} \)
41 \( 1 - 76.1T + 6.89e4T^{2} \)
43 \( 1 - 31.7iT - 7.95e4T^{2} \)
47 \( 1 + 60.8iT - 1.03e5T^{2} \)
53 \( 1 + 466. iT - 1.48e5T^{2} \)
59 \( 1 + 95.4T + 2.05e5T^{2} \)
61 \( 1 + 357.T + 2.26e5T^{2} \)
67 \( 1 + 87.8iT - 3.00e5T^{2} \)
71 \( 1 - 412.T + 3.57e5T^{2} \)
73 \( 1 + 331. iT - 3.89e5T^{2} \)
79 \( 1 - 248.T + 4.93e5T^{2} \)
83 \( 1 + 552. iT - 5.71e5T^{2} \)
89 \( 1 - 291.T + 7.04e5T^{2} \)
97 \( 1 + 198. iT - 9.12e5T^{2} \)
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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

−9.897604011548399245965494538967, −9.257141748036966567928332274900, −7.77025023345869879189363938765, −7.05348011539651302866322834037, −6.44723345872744177239921981291, −4.87136103419188486907960287166, −3.80462684146270974992960793031, −3.11443642796747224789452551097, −1.49515164926447225204229737772, −0.72051974713804041412184733083, 1.67466663559137733344811168422, 2.62355865962679078980611210546, 4.07085471921563623218222317302, 5.39316382378083743076235224012, 6.16441824988116594473007922306, 6.65945325298819485831993444231, 7.88128987362354710256959391132, 8.781839192474839391959795296788, 9.152157450056590178978807263527, 10.57491623442613981222060015002

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