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Properties

Label 2-675-45.4-c1-0-2
Degree $2$
Conductor $675$
Sign $-0.700 - 0.713i$
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  + (−2.28 + 1.31i)2-s + (2.46 − 4.27i)4-s + (−1.55 + 0.898i)7-s + 7.73i·8-s + (0.904 + 1.56i)11-s + (1.70 + 0.985i)13-s + (2.36 − 4.09i)14-s + (−5.24 − 9.08i)16-s − 4.80i·17-s − 2.96·19-s + (−4.12 − 2.38i)22-s + (1.50 + 0.866i)23-s − 5.19·26-s + 8.87i·28-s + (3.68 + 6.38i)29-s + ⋯
L(s)  = 1  + (−1.61 + 0.931i)2-s + (1.23 − 2.13i)4-s + (−0.588 + 0.339i)7-s + 2.73i·8-s + (0.272 + 0.472i)11-s + (0.473 + 0.273i)13-s + (0.632 − 1.09i)14-s + (−1.31 − 2.27i)16-s − 1.16i·17-s − 0.680·19-s + (−0.879 − 0.507i)22-s + (0.313 + 0.180i)23-s − 1.01·26-s + 1.67i·28-s + (0.684 + 1.18i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.700 - 0.713i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.700 - 0.713i$
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: \((2,\ 675,\ (\ :1/2),\ -0.700 - 0.713i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.192091 + 0.457794i\)
\(L(\frac12)\) \(\approx\) \(0.192091 + 0.457794i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.28 - 1.31i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (1.55 - 0.898i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.904 - 1.56i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.70 - 0.985i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.80iT - 17T^{2} \)
19 \( 1 + 2.96T + 19T^{2} \)
23 \( 1 + (-1.50 - 0.866i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.68 - 6.38i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.31 + 2.27i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 11.6iT - 37T^{2} \)
41 \( 1 + (1.23 - 2.13i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.30 + 3.63i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.44 - 3.14i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 1.72iT - 53T^{2} \)
59 \( 1 + (5.51 - 9.54i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.33 - 10.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.88 + 4.55i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.27T + 71T^{2} \)
73 \( 1 - 3.58iT - 73T^{2} \)
79 \( 1 + (-1.05 - 1.82i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.951 + 0.549i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 + (3.31 - 1.91i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.43393038834364018382807894987, −9.679834975424824402357341086697, −9.042934838834340836941155959306, −8.375105249189086257798479143126, −7.31129042898347019595156019469, −6.67685628322121384709310694499, −5.92051672637739943420302037014, −4.71712760700169063454510039871, −2.78972618913903698634204904040, −1.27548601749418006357902748948, 0.49468558151538656092042625016, 1.91385857497714822141929802078, 3.18460772642229542912598368008, 4.05453466672366983776266688699, 6.06919150234550240286587930602, 6.89187500414734023976723326846, 8.016806421775394170662230831676, 8.551578036054566641118469723851, 9.409707713502470099091278329884, 10.20567885298614795640404527961

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