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Properties

Label 1-513-513.221-r0-0-0
Degree $1$
Conductor $513$
Sign $-0.790 - 0.611i$
Analytic cond. $2.38236$
Root an. cond. $2.38236$
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  + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (−0.173 − 0.984i)5-s + (−0.939 + 0.342i)7-s + (−0.5 + 0.866i)8-s + (0.5 − 0.866i)10-s + (−0.766 − 0.642i)11-s + (−0.766 + 0.642i)13-s + (−0.939 − 0.342i)14-s + (−0.939 + 0.342i)16-s − 17-s + (0.939 − 0.342i)20-s + (−0.173 − 0.984i)22-s + (−0.173 − 0.984i)23-s + (−0.939 + 0.342i)25-s − 26-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (−0.173 − 0.984i)5-s + (−0.939 + 0.342i)7-s + (−0.5 + 0.866i)8-s + (0.5 − 0.866i)10-s + (−0.766 − 0.642i)11-s + (−0.766 + 0.642i)13-s + (−0.939 − 0.342i)14-s + (−0.939 + 0.342i)16-s − 17-s + (0.939 − 0.342i)20-s + (−0.173 − 0.984i)22-s + (−0.173 − 0.984i)23-s + (−0.939 + 0.342i)25-s − 26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.790 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.790 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $-0.790 - 0.611i$
Analytic conductor: \(2.38236\)
Root analytic conductor: \(2.38236\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{513} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 513,\ (0:\ ),\ -0.790 - 0.611i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03397648570 + 0.09945091274i\)
\(L(\frac12)\) \(\approx\) \(-0.03397648570 + 0.09945091274i\)
\(L(1)\) \(\approx\) \(0.8657093968 + 0.2982191750i\)
\(L(1)\) \(\approx\) \(0.8657093968 + 0.2982191750i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 + (-0.173 - 0.984i)T \)
7 \( 1 + (-0.939 + 0.342i)T \)
11 \( 1 + (-0.766 - 0.642i)T \)
13 \( 1 + (-0.766 + 0.642i)T \)
17 \( 1 - T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (-0.939 + 0.342i)T \)
31 \( 1 + (0.939 + 0.342i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.939 - 0.342i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.173 + 0.984i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (-0.766 + 0.642i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (-0.173 + 0.984i)T \)
89 \( 1 + T \)
97 \( 1 + (0.939 - 0.342i)T \)
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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

−22.96468528610043434487336629495, −22.25130918122226181778739519019, −21.6669901223529138844301518015, −20.41732232868620237512922502828, −19.81887002525408842947399013435, −19.06819355939564041797700564978, −18.2277572732837023754462506605, −17.273480531645016775807925731027, −15.69566585357753398402120822729, −15.38985961215844948239223044287, −14.42884008949112140212358810433, −13.37174839465193887231992062321, −12.88509476790551441489445226594, −11.784636080126297530513806524688, −10.910526337821452048659813621021, −10.06544567881507322289405136636, −9.56566799304091453400103293625, −7.71881332275682101040520856461, −6.88154556581052895249059239236, −6.01247475079660467472748185449, −4.87396333412696348316247303530, −3.76527870623892153106914420258, −2.90664932411696907545941035237, −2.0906782451949653106353329736, −0.03842245184610764837646265181, 2.21544558053684633168615399237, 3.2772808991453868707067212695, 4.447284393893914075009858990378, 5.174342737356140237676338199285, 6.20451804346026706538609625656, 7.03827403172578166913912352631, 8.28693961708604251326941158708, 8.85083437047019818789079794951, 9.99448371010822616274212726430, 11.47511257411732214582547578886, 12.21631132657090944024968643703, 13.12517592173175178543087250166, 13.49331493157638875109902407811, 14.814435510621978781041361861536, 15.63556451983170827010765589732, 16.408996804578386423647047534507, 16.805210311902540227460913508314, 18.012986747413164687586121450803, 19.10811850507518027409854997198, 20.05355841788766531858487206231, 20.87672966418182471473413452162, 21.77646723569906320250466742244, 22.37347384838514478206610895193, 23.39614757175691992861002664708, 24.19438704208960882308299314512

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