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Properties

Label 2-205-205.9-c1-0-16
Degree $2$
Conductor $205$
Sign $0.867 + 0.496i$
Analytic cond. $1.63693$
Root an. cond. $1.27942$
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  + 2.01·2-s + (1.65 − 1.65i)3-s + 2.04·4-s + (−2.11 + 0.728i)5-s + (3.32 − 3.32i)6-s + (−0.756 + 0.756i)7-s + 0.0829·8-s − 2.45i·9-s + (−4.24 + 1.46i)10-s + (1.97 + 1.97i)11-s + (3.37 − 3.37i)12-s + (0.895 − 0.895i)13-s + (−1.52 + 1.52i)14-s + (−2.28 + 4.69i)15-s − 3.91·16-s + (−0.688 − 0.688i)17-s + ⋯
L(s)  = 1  + 1.42·2-s + (0.953 − 0.953i)3-s + 1.02·4-s + (−0.945 + 0.325i)5-s + (1.35 − 1.35i)6-s + (−0.285 + 0.285i)7-s + 0.0293·8-s − 0.819i·9-s + (−1.34 + 0.463i)10-s + (0.596 + 0.596i)11-s + (0.973 − 0.973i)12-s + (0.248 − 0.248i)13-s + (−0.406 + 0.406i)14-s + (−0.590 + 1.21i)15-s − 0.978·16-s + (−0.166 − 0.166i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 + 0.496i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.867 + 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(205\)    =    \(5 \cdot 41\)
Sign: $0.867 + 0.496i$
Analytic conductor: \(1.63693\)
Root analytic conductor: \(1.27942\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{205} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 205,\ (\ :1/2),\ 0.867 + 0.496i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.48154 - 0.660036i\)
\(L(\frac12)\) \(\approx\) \(2.48154 - 0.660036i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.11 - 0.728i)T \)
41 \( 1 + (-5.94 + 2.38i)T \)
good2 \( 1 - 2.01T + 2T^{2} \)
3 \( 1 + (-1.65 + 1.65i)T - 3iT^{2} \)
7 \( 1 + (0.756 - 0.756i)T - 7iT^{2} \)
11 \( 1 + (-1.97 - 1.97i)T + 11iT^{2} \)
13 \( 1 + (-0.895 + 0.895i)T - 13iT^{2} \)
17 \( 1 + (0.688 + 0.688i)T + 17iT^{2} \)
19 \( 1 + (3.27 - 3.27i)T - 19iT^{2} \)
23 \( 1 + 6.02iT - 23T^{2} \)
29 \( 1 + (0.0412 + 0.0412i)T + 29iT^{2} \)
31 \( 1 + 2.91T + 31T^{2} \)
37 \( 1 - 5.82iT - 37T^{2} \)
43 \( 1 - 12.1T + 43T^{2} \)
47 \( 1 + (8.76 + 8.76i)T + 47iT^{2} \)
53 \( 1 + (-3.33 + 3.33i)T - 53iT^{2} \)
59 \( 1 + 1.23T + 59T^{2} \)
61 \( 1 + 8.72iT - 61T^{2} \)
67 \( 1 + (4.81 + 4.81i)T + 67iT^{2} \)
71 \( 1 + (4.73 + 4.73i)T + 71iT^{2} \)
73 \( 1 + 14.0T + 73T^{2} \)
79 \( 1 + (-7.36 - 7.36i)T + 79iT^{2} \)
83 \( 1 - 0.852iT - 83T^{2} \)
89 \( 1 + (-3.89 - 3.89i)T + 89iT^{2} \)
97 \( 1 + (-3.82 - 3.82i)T + 97iT^{2} \)
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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

−12.53037299933646337491411898732, −11.99665411628797885500130861130, −10.76137287601165604674035767240, −9.066670383660835970178330202987, −8.106617386763505950573109370622, −7.02877155226166559586814267737, −6.22357790390616132797163977471, −4.52017233117892676488237503843, −3.48020314596680585156361508259, −2.36557089091695798692551266599, 3.01700589597590079509188889641, 3.93092339045797498870433735475, 4.44757778290625960676307294102, 5.93553345133696773330037332740, 7.32220949777001581443009651596, 8.743110223357675971050707703473, 9.292057258966163563489848772053, 10.87692929780918286511113480401, 11.63916627292109169108851276367, 12.76395766185804007987855505837

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