Properties

Label 2-675-135.113-c1-0-50
Degree $2$
Conductor $675$
Sign $-0.813 + 0.581i$
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  + (1.75 + 0.153i)2-s + (−0.305 − 1.70i)3-s + (1.09 + 0.192i)4-s + (−0.274 − 3.04i)6-s + (−2.75 − 1.93i)7-s + (−1.51 − 0.406i)8-s + (−2.81 + 1.04i)9-s + (−1.11 + 3.07i)11-s + (−0.00542 − 1.92i)12-s + (−0.506 − 5.79i)13-s + (−4.55 − 3.81i)14-s + (−4.68 − 1.70i)16-s + (−1.11 + 0.298i)17-s + (−5.10 + 1.39i)18-s + (6.40 − 3.69i)19-s + ⋯
L(s)  = 1  + (1.24 + 0.108i)2-s + (−0.176 − 0.984i)3-s + (0.546 + 0.0963i)4-s + (−0.112 − 1.24i)6-s + (−1.04 − 0.730i)7-s + (−0.536 − 0.143i)8-s + (−0.937 + 0.347i)9-s + (−0.337 + 0.926i)11-s + (−0.00156 − 0.554i)12-s + (−0.140 − 1.60i)13-s + (−1.21 − 1.02i)14-s + (−1.17 − 0.426i)16-s + (−0.270 + 0.0723i)17-s + (−1.20 + 0.329i)18-s + (1.46 − 0.848i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 + 0.581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.813 + 0.581i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (518, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ -0.813 + 0.581i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.452324 - 1.41155i\)
\(L(\frac12)\) \(\approx\) \(0.452324 - 1.41155i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.305 + 1.70i)T \)
5 \( 1 \)
good2 \( 1 + (-1.75 - 0.153i)T + (1.96 + 0.347i)T^{2} \)
7 \( 1 + (2.75 + 1.93i)T + (2.39 + 6.57i)T^{2} \)
11 \( 1 + (1.11 - 3.07i)T + (-8.42 - 7.07i)T^{2} \)
13 \( 1 + (0.506 + 5.79i)T + (-12.8 + 2.25i)T^{2} \)
17 \( 1 + (1.11 - 0.298i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-6.40 + 3.69i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.09 + 1.56i)T + (-7.86 + 21.6i)T^{2} \)
29 \( 1 + (-0.381 + 0.320i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-0.573 + 3.25i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (-1.17 - 4.40i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.39 - 1.65i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (-1.68 - 3.62i)T + (-27.6 + 32.9i)T^{2} \)
47 \( 1 + (-2.84 + 4.06i)T + (-16.0 - 44.1i)T^{2} \)
53 \( 1 + (1.25 - 1.25i)T - 53iT^{2} \)
59 \( 1 + (-9.92 + 3.61i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (1.82 + 10.3i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-5.82 + 0.509i)T + (65.9 - 11.6i)T^{2} \)
71 \( 1 + (6.62 + 3.82i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.84 + 6.89i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.08 + 6.06i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (0.544 - 6.22i)T + (-81.7 - 14.4i)T^{2} \)
89 \( 1 + (-0.260 - 0.450i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.27 - 1.99i)T + (62.3 - 74.3i)T^{2} \)
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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

−10.20852165536430415030931245712, −9.435232621563387404956191914817, −8.035699702945236794237877046793, −7.20358403047006919460506955324, −6.51900059336825726615326153739, −5.58255353701468925857267863795, −4.78613739424959780291751227500, −3.40702887872189020937165414142, −2.64569267511291335928249305118, −0.51281078953382120786173097164, 2.65779106484953587432646873971, 3.49711749677580409374267617644, 4.27568065661386869186005841113, 5.49289581153332353920018081629, 5.84158799486882495500829682803, 6.90173159106956745912678195485, 8.599019353311877375968803960230, 9.231181709263564905462067149486, 9.915119675989944091541536932406, 11.13171630665047281226332117970

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