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Properties

Label 2-405-135.83-c1-0-0
Degree $2$
Conductor $405$
Sign $-0.989 + 0.145i$
Analytic cond. $3.23394$
Root an. cond. $1.79831$
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  + (−1.65 + 1.15i)2-s + (0.704 − 1.93i)4-s + (0.268 + 2.21i)5-s + (0.618 − 1.32i)7-s + (0.0310 + 0.115i)8-s + (−3.00 − 3.35i)10-s + (−1.86 + 2.21i)11-s + (−3.51 + 5.02i)13-s + (0.512 + 2.90i)14-s + (2.97 + 2.49i)16-s + (0.833 − 3.10i)17-s + (3.51 + 2.03i)19-s + (4.48 + 1.04i)20-s + (0.508 − 5.80i)22-s + (−4.58 + 2.14i)23-s + ⋯
L(s)  = 1  + (−1.16 + 0.817i)2-s + (0.352 − 0.967i)4-s + (0.120 + 0.992i)5-s + (0.233 − 0.501i)7-s + (0.0109 + 0.0409i)8-s + (−0.951 − 1.06i)10-s + (−0.560 + 0.668i)11-s + (−0.975 + 1.39i)13-s + (0.136 + 0.776i)14-s + (0.742 + 0.623i)16-s + (0.202 − 0.753i)17-s + (0.806 + 0.465i)19-s + (1.00 + 0.233i)20-s + (0.108 − 1.23i)22-s + (−0.957 + 0.446i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.145i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.989 + 0.145i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ -0.989 + 0.145i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0319389 - 0.437863i\)
\(L(\frac12)\) \(\approx\) \(0.0319389 - 0.437863i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.268 - 2.21i)T \)
good2 \( 1 + (1.65 - 1.15i)T + (0.684 - 1.87i)T^{2} \)
7 \( 1 + (-0.618 + 1.32i)T + (-4.49 - 5.36i)T^{2} \)
11 \( 1 + (1.86 - 2.21i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (3.51 - 5.02i)T + (-4.44 - 12.2i)T^{2} \)
17 \( 1 + (-0.833 + 3.10i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-3.51 - 2.03i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.58 - 2.14i)T + (14.7 - 17.6i)T^{2} \)
29 \( 1 + (0.368 - 2.08i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (3.20 + 1.16i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (11.3 + 3.04i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.839 - 0.147i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (0.0641 + 0.732i)T + (-42.3 + 7.46i)T^{2} \)
47 \( 1 + (2.61 + 1.21i)T + (30.2 + 36.0i)T^{2} \)
53 \( 1 + (-0.0757 + 0.0757i)T - 53iT^{2} \)
59 \( 1 + (-2.89 + 2.43i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (4.21 - 1.53i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-3.65 - 2.55i)T + (22.9 + 62.9i)T^{2} \)
71 \( 1 + (-7.84 + 4.53i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (8.04 - 2.15i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-1.44 - 0.254i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (-9.69 - 13.8i)T + (-28.3 + 77.9i)T^{2} \)
89 \( 1 + (3.05 - 5.28i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-16.1 + 1.41i)T + (95.5 - 16.8i)T^{2} \)
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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

−11.51281849932363451930863773941, −10.37420996491271573436730177789, −9.842575274211322582018552017364, −9.086688074423638535685030923041, −7.64506772158002721986605758834, −7.35681002105533185966491854961, −6.56958536611748727985331428903, −5.23749960536582014368138098676, −3.71558397919037740708706520934, −1.99177792850579375813037325540, 0.40224370568395652211551868479, 1.94361646514324600142641366009, 3.20043971255025679460830107644, 5.07309929381557275018339586035, 5.71466252316868071607663540218, 7.65163042617805783906505503139, 8.300294303225093726524358843021, 8.948906844722794208593664511131, 9.995830771145236325616453252293, 10.47176628619153884260689529538

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