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Properties

Label 2-205-205.9-c1-0-14
Degree $2$
Conductor $205$
Sign $-0.0660 + 0.997i$
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  + 0.160·2-s + (0.887 − 0.887i)3-s − 1.97·4-s + (−1.87 − 1.21i)5-s + (0.142 − 0.142i)6-s + (3.03 − 3.03i)7-s − 0.636·8-s + 1.42i·9-s + (−0.301 − 0.194i)10-s + (−3.91 − 3.91i)11-s + (−1.75 + 1.75i)12-s + (3.92 − 3.92i)13-s + (0.487 − 0.487i)14-s + (−2.74 + 0.592i)15-s + 3.84·16-s + (−0.489 − 0.489i)17-s + ⋯
L(s)  = 1  + 0.113·2-s + (0.512 − 0.512i)3-s − 0.987·4-s + (−0.840 − 0.541i)5-s + (0.0580 − 0.0580i)6-s + (1.14 − 1.14i)7-s − 0.225·8-s + 0.474i·9-s + (−0.0952 − 0.0614i)10-s + (−1.17 − 1.17i)11-s + (−0.506 + 0.506i)12-s + (1.08 − 1.08i)13-s + (0.130 − 0.130i)14-s + (−0.708 + 0.153i)15-s + 0.961·16-s + (−0.118 − 0.118i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0660 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0660 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(205\)    =    \(5 \cdot 41\)
Sign: $-0.0660 + 0.997i$
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: \((2,\ 205,\ (\ :1/2),\ -0.0660 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.732189 - 0.782220i\)
\(L(\frac12)\) \(\approx\) \(0.732189 - 0.782220i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.87 + 1.21i)T \)
41 \( 1 + (-6.31 + 1.07i)T \)
good2 \( 1 - 0.160T + 2T^{2} \)
3 \( 1 + (-0.887 + 0.887i)T - 3iT^{2} \)
7 \( 1 + (-3.03 + 3.03i)T - 7iT^{2} \)
11 \( 1 + (3.91 + 3.91i)T + 11iT^{2} \)
13 \( 1 + (-3.92 + 3.92i)T - 13iT^{2} \)
17 \( 1 + (0.489 + 0.489i)T + 17iT^{2} \)
19 \( 1 + (4.11 - 4.11i)T - 19iT^{2} \)
23 \( 1 - 2.05iT - 23T^{2} \)
29 \( 1 + (-3.97 - 3.97i)T + 29iT^{2} \)
31 \( 1 - 4.84T + 31T^{2} \)
37 \( 1 + 5.48iT - 37T^{2} \)
43 \( 1 + 2.20T + 43T^{2} \)
47 \( 1 + (-0.453 - 0.453i)T + 47iT^{2} \)
53 \( 1 + (0.168 - 0.168i)T - 53iT^{2} \)
59 \( 1 + 3.84T + 59T^{2} \)
61 \( 1 - 5.66iT - 61T^{2} \)
67 \( 1 + (1.99 + 1.99i)T + 67iT^{2} \)
71 \( 1 + (1.69 + 1.69i)T + 71iT^{2} \)
73 \( 1 - 15.0T + 73T^{2} \)
79 \( 1 + (-0.167 - 0.167i)T + 79iT^{2} \)
83 \( 1 + 8.51iT - 83T^{2} \)
89 \( 1 + (1.65 + 1.65i)T + 89iT^{2} \)
97 \( 1 + (-3.53 - 3.53i)T + 97iT^{2} \)
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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

−12.50662399183128136808908601222, −10.92597577596501100007733824447, −10.56514893340628076068685879605, −8.587046531552647651784837353503, −8.163195112812793542560715997887, −7.68098136196555629541685492728, −5.56638920645668943801070465386, −4.54578109223245017631050787751, −3.41504747631545633267292414542, −0.955327051798723041369677353443, 2.60151445197573007582978034916, 4.22735750094277823392236776538, 4.77954977768258656711243779543, 6.45976399623157119616754027139, 8.127449767542272765763930075079, 8.557008791589398900986118051352, 9.551079903012666658864726136882, 10.72494114322067096103028644969, 11.76573607964832743439785250520, 12.57687660231037359344392136830

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