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Properties

Label 1-7e2-49.13-r1-0-0
Degree $1$
Conductor $49$
Sign $-0.648 - 0.761i$
Analytic cond. $5.26578$
Root an. cond. $5.26578$
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  + (−0.900 + 0.433i)2-s + (0.222 − 0.974i)3-s + (0.623 − 0.781i)4-s + (0.222 − 0.974i)5-s + (0.222 + 0.974i)6-s + (−0.222 + 0.974i)8-s + (−0.900 − 0.433i)9-s + (0.222 + 0.974i)10-s + (−0.900 + 0.433i)11-s + (−0.623 − 0.781i)12-s + (0.900 − 0.433i)13-s + (−0.900 − 0.433i)15-s + (−0.222 − 0.974i)16-s + (−0.623 − 0.781i)17-s + 18-s − 19-s + ⋯
L(s)  = 1  + (−0.900 + 0.433i)2-s + (0.222 − 0.974i)3-s + (0.623 − 0.781i)4-s + (0.222 − 0.974i)5-s + (0.222 + 0.974i)6-s + (−0.222 + 0.974i)8-s + (−0.900 − 0.433i)9-s + (0.222 + 0.974i)10-s + (−0.900 + 0.433i)11-s + (−0.623 − 0.781i)12-s + (0.900 − 0.433i)13-s + (−0.900 − 0.433i)15-s + (−0.222 − 0.974i)16-s + (−0.623 − 0.781i)17-s + 18-s − 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.648 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.648 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.648 - 0.761i$
Analytic conductor: \(5.26578\)
Root analytic conductor: \(5.26578\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 49,\ (1:\ ),\ -0.648 - 0.761i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3430338783 - 0.7425322474i\)
\(L(\frac12)\) \(\approx\) \(0.3430338783 - 0.7425322474i\)
\(L(1)\) \(\approx\) \(0.6381898100 - 0.3329432151i\)
\(L(1)\) \(\approx\) \(0.6381898100 - 0.3329432151i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 + (-0.900 + 0.433i)T \)
3 \( 1 + (0.222 - 0.974i)T \)
5 \( 1 + (0.222 - 0.974i)T \)
11 \( 1 + (-0.900 + 0.433i)T \)
13 \( 1 + (0.900 - 0.433i)T \)
17 \( 1 + (-0.623 - 0.781i)T \)
19 \( 1 - T \)
23 \( 1 + (0.623 - 0.781i)T \)
29 \( 1 + (0.623 + 0.781i)T \)
31 \( 1 - T \)
37 \( 1 + (0.623 + 0.781i)T \)
41 \( 1 + (0.222 - 0.974i)T \)
43 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (0.900 - 0.433i)T \)
53 \( 1 + (0.623 - 0.781i)T \)
59 \( 1 + (0.222 + 0.974i)T \)
61 \( 1 + (-0.623 - 0.781i)T \)
67 \( 1 + T \)
71 \( 1 + (0.623 - 0.781i)T \)
73 \( 1 + (0.900 + 0.433i)T \)
79 \( 1 + T \)
83 \( 1 + (0.900 + 0.433i)T \)
89 \( 1 + (0.900 + 0.433i)T \)
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

−33.98836440294976044649814766948, −33.17012068403343072753103560118, −31.4456605407810626394234552898, −30.45814844011693481021889476712, −29.12325160225205358370548240480, −28.12129119324467179153034054961, −26.8859720372463498207033002235, −26.19104724397955325874576107632, −25.37862981546304768575928168896, −23.26883772544913077802971689232, −21.63830618840411725815625821403, −21.23512866805905211479305598942, −19.70407177138569428942382907089, −18.63016350166843136059947531571, −17.38213803410187116352609075318, −16.042946689603258902231496815826, −15.02067574958208697591472212300, −13.33379424749684009460849082121, −11.13461553563507124338930876297, −10.654754976158458194893196228553, −9.29711190686766062363123164485, −8.028222522368935241349244935339, −6.20955800913001390301304853844, −3.80439418237328399958543743528, −2.450817573925278222369971434790, 0.60056917268828823195767373265, 2.20600916638274366885652660232, 5.3273235709835300239371459186, 6.77759266732586358806650115353, 8.177389398440013871384527971223, 9.01461692871406346610692369953, 10.77939776993237889842398255019, 12.46071179133927016620165387661, 13.609956633092308939403326271100, 15.27475485215692547402810999317, 16.59544110489997644504699280495, 17.78711495775712956310234342298, 18.60833122611105934859990377817, 20.0438257629569380330635635655, 20.743979448449691130826902338024, 23.24359749257749210998054267726, 24.06053618772629862671926060270, 25.18996029158326816353397464534, 25.80782201065911640541746108261, 27.439005243846592102023208979520, 28.65469980739025150032462810622, 29.248852173955944215190399537594, 30.786828813113829115214818548790, 32.08792581785433304386675529324, 33.308707025842836179157827816272

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