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Properties

Label 2-205-205.9-c1-0-1
Degree $2$
Conductor $205$
Sign $-0.898 - 0.438i$
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  + 1.22·2-s + (−1.56 + 1.56i)3-s − 0.503·4-s + (−2.19 + 0.434i)5-s + (−1.90 + 1.90i)6-s + (−0.202 + 0.202i)7-s − 3.06·8-s − 1.86i·9-s + (−2.68 + 0.532i)10-s + (0.118 + 0.118i)11-s + (0.785 − 0.785i)12-s + (−1.76 + 1.76i)13-s + (−0.248 + 0.248i)14-s + (2.74 − 4.10i)15-s − 2.73·16-s + (3.70 + 3.70i)17-s + ⋯
L(s)  = 1  + 0.865·2-s + (−0.900 + 0.900i)3-s − 0.251·4-s + (−0.980 + 0.194i)5-s + (−0.779 + 0.779i)6-s + (−0.0767 + 0.0767i)7-s − 1.08·8-s − 0.623i·9-s + (−0.848 + 0.168i)10-s + (0.0356 + 0.0356i)11-s + (0.226 − 0.226i)12-s + (−0.489 + 0.489i)13-s + (−0.0663 + 0.0663i)14-s + (0.708 − 1.05i)15-s − 0.684·16-s + (0.898 + 0.898i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 - 0.438i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.898 - 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(205\)    =    \(5 \cdot 41\)
Sign: $-0.898 - 0.438i$
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: \((2,\ 205,\ (\ :1/2),\ -0.898 - 0.438i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.140232 + 0.607630i\)
\(L(\frac12)\) \(\approx\) \(0.140232 + 0.607630i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.19 - 0.434i)T \)
41 \( 1 + (5.34 - 3.53i)T \)
good2 \( 1 - 1.22T + 2T^{2} \)
3 \( 1 + (1.56 - 1.56i)T - 3iT^{2} \)
7 \( 1 + (0.202 - 0.202i)T - 7iT^{2} \)
11 \( 1 + (-0.118 - 0.118i)T + 11iT^{2} \)
13 \( 1 + (1.76 - 1.76i)T - 13iT^{2} \)
17 \( 1 + (-3.70 - 3.70i)T + 17iT^{2} \)
19 \( 1 + (0.594 - 0.594i)T - 19iT^{2} \)
23 \( 1 - 3.99iT - 23T^{2} \)
29 \( 1 + (-2.50 - 2.50i)T + 29iT^{2} \)
31 \( 1 + 1.73T + 31T^{2} \)
37 \( 1 + 5.93iT - 37T^{2} \)
43 \( 1 + 3.38T + 43T^{2} \)
47 \( 1 + (-4.17 - 4.17i)T + 47iT^{2} \)
53 \( 1 + (-6.15 + 6.15i)T - 53iT^{2} \)
59 \( 1 + 13.0T + 59T^{2} \)
61 \( 1 - 6.82iT - 61T^{2} \)
67 \( 1 + (10.7 + 10.7i)T + 67iT^{2} \)
71 \( 1 + (1.46 + 1.46i)T + 71iT^{2} \)
73 \( 1 - 3.95T + 73T^{2} \)
79 \( 1 + (0.412 + 0.412i)T + 79iT^{2} \)
83 \( 1 - 13.3iT - 83T^{2} \)
89 \( 1 + (-8.68 - 8.68i)T + 89iT^{2} \)
97 \( 1 + (0.699 + 0.699i)T + 97iT^{2} \)
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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

−12.51847805046560586489107270886, −11.99518167014707803472683038644, −11.07755091078782303477856000585, −10.12151066216770951674510829378, −9.029899350914701040169513263178, −7.70651981220995323530564165781, −6.22975785442146071689383204417, −5.19830766201810032656473297082, −4.31708263422188152898156128016, −3.42372350715637961184159255158, 0.46033172665842631963707317695, 3.20407272768036826856167465180, 4.61547304789627778740458707715, 5.52913889047003153104047373472, 6.69631958611399213538457216708, 7.67174969346904490543482510598, 8.851742554111022493839028794891, 10.28977047152721386840462488807, 11.74328135684224435102607772175, 12.00733082846671135578095104797

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