Properties

Label 1-709-709.203-r0-0-0
Degree $1$
Conductor $709$
Sign $0.0966 - 0.995i$
Analytic cond. $3.29258$
Root an. cond. $3.29258$
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  + (0.484 + 0.874i)2-s + (0.574 + 0.818i)3-s + (−0.530 + 0.847i)4-s + (−0.887 − 0.461i)5-s + (−0.437 + 0.899i)6-s + (0.388 − 0.921i)7-s + (−0.998 − 0.0532i)8-s + (−0.339 + 0.940i)9-s + (−0.0266 − 0.999i)10-s + (−0.931 − 0.364i)11-s + (−0.998 + 0.0532i)12-s + (0.388 + 0.921i)13-s + (0.994 − 0.106i)14-s + (−0.132 − 0.991i)15-s + (−0.437 − 0.899i)16-s + (−0.931 + 0.364i)17-s + ⋯
L(s)  = 1  + (0.484 + 0.874i)2-s + (0.574 + 0.818i)3-s + (−0.530 + 0.847i)4-s + (−0.887 − 0.461i)5-s + (−0.437 + 0.899i)6-s + (0.388 − 0.921i)7-s + (−0.998 − 0.0532i)8-s + (−0.339 + 0.940i)9-s + (−0.0266 − 0.999i)10-s + (−0.931 − 0.364i)11-s + (−0.998 + 0.0532i)12-s + (0.388 + 0.921i)13-s + (0.994 − 0.106i)14-s + (−0.132 − 0.991i)15-s + (−0.437 − 0.899i)16-s + (−0.931 + 0.364i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0966 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0966 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.0966 - 0.995i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ 0.0966 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03150755668 + 0.02859511120i\)
\(L(\frac12)\) \(\approx\) \(-0.03150755668 + 0.02859511120i\)
\(L(1)\) \(\approx\) \(0.7229829729 + 0.5437817782i\)
\(L(1)\) \(\approx\) \(0.7229829729 + 0.5437817782i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (0.484 + 0.874i)T \)
3 \( 1 + (0.574 + 0.818i)T \)
5 \( 1 + (-0.887 - 0.461i)T \)
7 \( 1 + (0.388 - 0.921i)T \)
11 \( 1 + (-0.931 - 0.364i)T \)
13 \( 1 + (0.388 + 0.921i)T \)
17 \( 1 + (-0.931 + 0.364i)T \)
19 \( 1 + (-0.998 - 0.0532i)T \)
23 \( 1 + (-0.769 - 0.638i)T \)
29 \( 1 + (-0.987 - 0.159i)T \)
31 \( 1 + (-0.132 - 0.991i)T \)
37 \( 1 + (0.388 - 0.921i)T \)
41 \( 1 + (0.949 + 0.314i)T \)
43 \( 1 + (-0.237 + 0.971i)T \)
47 \( 1 + (-0.769 - 0.638i)T \)
53 \( 1 + (-0.132 - 0.991i)T \)
59 \( 1 + (-0.998 - 0.0532i)T \)
61 \( 1 + (0.734 + 0.678i)T \)
67 \( 1 + (-0.833 - 0.552i)T \)
71 \( 1 + (-0.0266 + 0.999i)T \)
73 \( 1 + (0.185 + 0.982i)T \)
79 \( 1 + (0.658 - 0.752i)T \)
83 \( 1 + (-0.697 + 0.716i)T \)
89 \( 1 + (-0.617 - 0.786i)T \)
97 \( 1 + (0.861 + 0.507i)T \)
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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

−22.027627687180867409655363346785, −20.98634905344625182967475104668, −20.2974167082746076920562357847, −19.62505716116546895648703023449, −18.81563221403775953123526734846, −18.20886836152107140835629524324, −17.729685105328803754215073072085, −15.63321940199951863000730173974, −15.28068402026965784526037294548, −14.529600087676546569859619192359, −13.51357441044332638961594506195, −12.72760747839983106007965052985, −12.164997347885826868369955341793, −11.25327758769588092650056369392, −10.561481662452004239481597051295, −9.25888021771695046682809294459, −8.393464867586104555741307925453, −7.7013613660034706554165683605, −6.472970840896473898808462675290, −5.51558269048623018058229688579, −4.3843884698098360442401241747, −3.256200655413756965896808094569, −2.574936380788442234537141222370, −1.705547629055896861930508814048, −0.01483160962383213488476970361, 2.27735447633492278463112308963, 3.70481440246791903711926613900, 4.21493666112323457031031320764, 4.781695980195174186543954883409, 6.03301182299260055539710473387, 7.24933254240837060384066955956, 8.09494937626783184271784289418, 8.528690744765704961964369847113, 9.54630534719843316823824300822, 10.870311508005273069906566825790, 11.41512680666098427841290542972, 12.94118793414555707207582171513, 13.36445885723530908108591997453, 14.44356188929279605262611086868, 14.98686650430010787891844979953, 15.94243886630537988523974461937, 16.399034097372871673470992579532, 17.05692811713771265526484670185, 18.26135640680272471461510869829, 19.3158370539689206695142551182, 20.19420190288322225024158577578, 20.98527329178524297595426689885, 21.47098909678126930300532459268, 22.60233037588667789044797401396, 23.32232413582756940647466886726

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