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Properties

Label 2-675-27.22-c1-0-49
Degree $2$
Conductor $675$
Sign $-0.653 + 0.757i$
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  + (1.62 − 1.36i)2-s + (0.986 − 1.42i)3-s + (0.430 − 2.44i)4-s + (−0.338 − 3.65i)6-s + (−0.168 − 0.957i)7-s + (−0.508 − 0.880i)8-s + (−1.05 − 2.80i)9-s + (0.297 + 0.108i)11-s + (−3.05 − 3.02i)12-s + (1.15 + 0.973i)13-s + (−1.57 − 1.32i)14-s + (2.63 + 0.960i)16-s + (0.587 − 1.01i)17-s + (−5.53 − 3.11i)18-s + (−3.11 − 5.38i)19-s + ⋯
L(s)  = 1  + (1.14 − 0.962i)2-s + (0.569 − 0.822i)3-s + (0.215 − 1.22i)4-s + (−0.138 − 1.49i)6-s + (−0.0638 − 0.361i)7-s + (−0.179 − 0.311i)8-s + (−0.351 − 0.936i)9-s + (0.0897 + 0.0326i)11-s + (−0.881 − 0.872i)12-s + (0.321 + 0.269i)13-s + (−0.421 − 0.353i)14-s + (0.659 + 0.240i)16-s + (0.142 − 0.246i)17-s + (−1.30 − 0.734i)18-s + (−0.713 − 1.23i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.653 + 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.653 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.653 + 0.757i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ -0.653 + 0.757i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34257 - 2.93136i\)
\(L(\frac12)\) \(\approx\) \(1.34257 - 2.93136i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.986 + 1.42i)T \)
5 \( 1 \)
good2 \( 1 + (-1.62 + 1.36i)T + (0.347 - 1.96i)T^{2} \)
7 \( 1 + (0.168 + 0.957i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (-0.297 - 0.108i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (-1.15 - 0.973i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.587 + 1.01i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.11 + 5.38i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.375 - 2.12i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (3.37 - 2.83i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (1.50 - 8.54i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (2.23 - 3.86i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.47 - 3.75i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-5.25 - 1.91i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.429 - 2.43i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 - 10.8T + 53T^{2} \)
59 \( 1 + (-1.62 + 0.589i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-0.176 - 0.999i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (0.656 + 0.550i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-4.79 + 8.31i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.62 + 13.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.59 - 7.20i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (3.58 - 3.01i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (-7.74 - 13.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.21 + 1.89i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.56417314625224050394968606236, −9.347081716866319151250948893997, −8.526923685343059574219119768890, −7.39206851751423087915316630039, −6.57905370223547202702574396983, −5.46063028422900635968733143240, −4.33318864812752076962702499288, −3.38940474717958361661909244113, −2.47490457043454455482152803912, −1.29383523158420305280958281267, 2.41133679334654022614194492814, 3.83686296053500281556186554679, 4.17522561281190791329402082436, 5.60810353140733925215839468013, 5.87303197272615699740979666667, 7.26631143956555815013489155919, 8.075115898910563730809162840868, 8.895941542968048440983080993922, 9.956369464671590993054189384578, 10.68921980319224206238842714027

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