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Properties

Label 1-205-205.163-r1-0-0
Degree $1$
Conductor $205$
Sign $0.850 - 0.525i$
Analytic cond. $22.0303$
Root an. cond. $22.0303$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s − i·3-s − 4-s − 6-s + i·7-s + i·8-s − 9-s − 11-s + i·12-s − i·13-s + 14-s + 16-s + i·17-s + i·18-s + 19-s + ⋯
L(s)  = 1  − i·2-s − i·3-s − 4-s − 6-s + i·7-s + i·8-s − 9-s − 11-s + i·12-s − i·13-s + 14-s + 16-s + i·17-s + i·18-s + 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(205\)    =    \(5 \cdot 41\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(22.0303\)
Root analytic conductor: \(22.0303\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{205} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 205,\ (1:\ ),\ 0.850 - 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.095908123 - 0.3113245318i\)
\(L(\frac12)\) \(\approx\) \(1.095908123 - 0.3113245318i\)
\(L(1)\) \(\approx\) \(0.7465935824 - 0.4614202097i\)
\(L(1)\) \(\approx\) \(0.7465935824 - 0.4614202097i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
41 \( 1 \)
good2 \( 1 \)
3 \( 1 + T \)
7 \( 1 - iT \)
11 \( 1 - T \)
13 \( 1 \)
17 \( 1 - T \)
19 \( 1 + iT \)
23 \( 1 + iT \)
29 \( 1 - T \)
31 \( 1 \)
37 \( 1 - T \)
43 \( 1 - iT \)
47 \( 1 + T \)
53 \( 1 \)
59 \( 1 + T \)
61 \( 1 + iT \)
67 \( 1 + iT \)
71 \( 1 + T \)
73 \( 1 \)
79 \( 1 + T \)
83 \( 1 + iT \)
89 \( 1 + iT \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−26.62897564732045501993985590583, −26.00739236075160844459219400020, −24.836077782189050590325757811334, −23.70313879239381262015736490808, −23.02598149650652156492727267243, −22.14000289016854559110482498895, −21.049752539089481775789766993545, −20.22402890557881065089826596186, −18.822758437429679778521010646857, −17.760099877149118742261036799060, −16.699218157799549997629026605, −16.15263041894443552211021828613, −15.286070458547336426731641284617, −14.010704539330231127721632657521, −13.6684699277569308034348531445, −11.92336872468279340519917696265, −10.50526374568294951558060386360, −9.78126845670915041776242480634, −8.671640142098028704120461654072, −7.56638239962088160776962322386, −6.46610183133596431407422053596, −5.03606488693049776012805863612, −4.43048570582639836091693804478, −3.09694565880848346687487784006, −0.47178790817082617212729109865, 1.07073859483187670491603280330, 2.39312738644426653769592753034, 3.20231782593223669679666890198, 5.157249734311786152537059091101, 5.9431394803103944759824253283, 7.79245316164298767361414846129, 8.440887064870969268823262208484, 9.73743959748473481480906435956, 10.92953258863418014207218993070, 11.94689170141306093918872892567, 12.722693317408852705056360601634, 13.41992548927036334401448133429, 14.600697860967637123940038658879, 15.787800415046097469435224572, 17.66034322930793068582763108549, 17.905797533143626054980532210763, 19.0065142869150050926994176845, 19.6389587957253970879888295395, 20.74466395709857404320061726989, 21.68286731297503794533622738650, 22.706443545894191996023560026503, 23.499873994861999987363917506704, 24.54033843112304869558072857396, 25.54150907226362070797140435956, 26.507855769621251139323548938057

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