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Properties

Label 1-2900-2900.2279-r0-0-0
Degree $1$
Conductor $2900$
Sign $0.843 - 0.537i$
Analytic cond. $13.4675$
Root an. cond. $13.4675$
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  + (0.951 − 0.309i)3-s + 7-s + (0.809 − 0.587i)9-s + (0.587 − 0.809i)11-s + (−0.809 + 0.587i)13-s + (0.951 + 0.309i)17-s + (0.951 + 0.309i)19-s + (0.951 − 0.309i)21-s + (−0.809 − 0.587i)23-s + (0.587 − 0.809i)27-s + (0.951 + 0.309i)31-s + (0.309 − 0.951i)33-s + (0.587 + 0.809i)37-s + (−0.587 + 0.809i)39-s + (−0.587 − 0.809i)41-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)3-s + 7-s + (0.809 − 0.587i)9-s + (0.587 − 0.809i)11-s + (−0.809 + 0.587i)13-s + (0.951 + 0.309i)17-s + (0.951 + 0.309i)19-s + (0.951 − 0.309i)21-s + (−0.809 − 0.587i)23-s + (0.587 − 0.809i)27-s + (0.951 + 0.309i)31-s + (0.309 − 0.951i)33-s + (0.587 + 0.809i)37-s + (−0.587 + 0.809i)39-s + (−0.587 − 0.809i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.843 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.843 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2900\)    =    \(2^{2} \cdot 5^{2} \cdot 29\)
Sign: $0.843 - 0.537i$
Analytic conductor: \(13.4675\)
Root analytic conductor: \(13.4675\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2900} (2279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2900,\ (0:\ ),\ 0.843 - 0.537i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.028806335 - 0.8828367496i\)
\(L(\frac12)\) \(\approx\) \(3.028806335 - 0.8828367496i\)
\(L(1)\) \(\approx\) \(1.751952073 - 0.2718198882i\)
\(L(1)\) \(\approx\) \(1.751952073 - 0.2718198882i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
29 \( 1 \)
good3 \( 1 + (0.951 - 0.309i)T \)
7 \( 1 + T \)
11 \( 1 + (0.587 - 0.809i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
17 \( 1 + (0.951 + 0.309i)T \)
19 \( 1 + (0.951 + 0.309i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (0.951 + 0.309i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (-0.587 - 0.809i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.951 + 0.309i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (-0.587 + 0.809i)T \)
67 \( 1 + (-0.309 + 0.951i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.587 + 0.809i)T \)
79 \( 1 + (-0.951 + 0.309i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (-0.587 + 0.809i)T \)
97 \( 1 + (0.951 - 0.309i)T \)
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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

−19.53769478359223418241657858672, −18.35704850257889048407565639987, −17.95263296220724504647180885936, −17.12367978865856603688305191324, −16.35134597358018577451011140681, −15.442554159956620267805559801003, −14.93815965587602726022738117262, −14.32064595281054247860494690999, −13.82858777151396678049836992152, −12.901724978859602400561151937877, −12.04256966145037054357886398114, −11.51881449086509378588351862825, −10.41680489023553235326755614205, −9.74293967070555279001195984917, −9.34399462096341557401782436505, −8.240210404327104705803967661926, −7.677047020927905463501268420441, −7.28217370056732330597355006899, −6.00285669217167872982089197865, −4.92110962714315473403136324285, −4.6049819065676273978093426138, −3.54970905815312109972683800005, −2.79254706549290495123461532563, −1.905081784638137994110340197582, −1.146826520362716858683180918597, 0.995875888611480006935858829842, 1.68232123441580324378275794238, 2.568901183983663112207346718366, 3.44567500562135917635843421539, 4.18191622203265600043335747160, 5.04036861481254938380904373607, 5.98793663380452171065617744609, 6.9084433258389768011574359626, 7.61661961781653406113361798967, 8.33684870452389615502855465983, 8.75415163553877955958306625892, 9.83874811350012567878378077860, 10.22025512762483243393776949533, 11.63140952875023261264717248815, 11.80748100090790705618385960873, 12.72247940809775857781313337433, 13.71838520514536635380335999280, 14.32832641090866020751929054431, 14.45646770478775919897163367893, 15.433911458779940404760324420582, 16.27653910162912652225388275779, 17.0038022023756746600660219480, 17.75775214469517287710708573555, 18.57367135747336429368238841295, 19.045598607728953815947324523804

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