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Properties

Label 1-205-205.122-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: $\chi_{205} (122, \cdot )$
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

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

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