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Properties

Label 2-675-9.4-c1-0-2
Degree $2$
Conductor $675$
Sign $0.999 + 0.0290i$
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  + (−0.736 − 1.27i)2-s + (−0.0852 + 0.147i)4-s + (1.93 + 3.34i)7-s − 2.69·8-s + (0.130 + 0.225i)11-s + (−2.03 + 3.53i)13-s + (2.84 − 4.93i)14-s + (2.15 + 3.73i)16-s + 3.26·17-s + 4.24·19-s + (0.191 − 0.332i)22-s + (−4.34 + 7.53i)23-s + 6.00·26-s − 0.659·28-s + (2.11 + 3.65i)29-s + ⋯
L(s)  = 1  + (−0.520 − 0.902i)2-s + (−0.0426 + 0.0738i)4-s + (0.730 + 1.26i)7-s − 0.952·8-s + (0.0392 + 0.0679i)11-s + (−0.565 + 0.979i)13-s + (0.761 − 1.31i)14-s + (0.538 + 0.933i)16-s + 0.790·17-s + 0.974·19-s + (0.0408 − 0.0708i)22-s + (−0.906 + 1.57i)23-s + 1.17·26-s − 0.124·28-s + (0.392 + 0.678i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0290i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.999 + 0.0290i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.999 + 0.0290i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13012 - 0.0164065i\)
\(L(\frac12)\) \(\approx\) \(1.13012 - 0.0164065i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.736 + 1.27i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-1.93 - 3.34i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.130 - 0.225i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.03 - 3.53i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.26T + 17T^{2} \)
19 \( 1 - 4.24T + 19T^{2} \)
23 \( 1 + (4.34 - 7.53i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.11 - 3.65i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.32 - 2.29i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.27T + 37T^{2} \)
41 \( 1 + (-2.82 + 4.88i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.53 + 7.85i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.714 - 1.23i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 11.3T + 53T^{2} \)
59 \( 1 + (3.56 - 6.16i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.26 + 2.18i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.64 + 9.77i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.38T + 71T^{2} \)
73 \( 1 + 0.403T + 73T^{2} \)
79 \( 1 + (-1.52 - 2.63i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.29 + 3.96i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 7.17T + 89T^{2} \)
97 \( 1 + (-1.55 - 2.69i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.48491197786206699942295071953, −9.555360037845783584835422524362, −9.116854396614166909910550509426, −8.155967134474683375486981463801, −7.09616074240583852579511718686, −5.77004675810904433297161822468, −5.18708536622883025262239400577, −3.59100762900733806563108325083, −2.38021449095739538253880278496, −1.50793206761887389993996670212, 0.76114127874463738391218017558, 2.78990631505765155350396978147, 4.06406061164483134410964251451, 5.21898523657193234738475139457, 6.24797617808382680421805541817, 7.23845298445667041183423705519, 7.889145856122304893464464939883, 8.308468620833779506700511668232, 9.705088349591845247964037310053, 10.22611550803475971545591553828

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