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Properties

Label 1-55e2-3025.413-r0-0-0
Degree $1$
Conductor $3025$
Sign $0.847 - 0.530i$
Analytic cond. $14.0480$
Root an. cond. $14.0480$
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  + (−0.336 + 0.941i)2-s + (0.587 − 0.809i)3-s + (−0.774 − 0.633i)4-s + (0.564 + 0.825i)6-s + (0.113 + 0.993i)7-s + (0.856 − 0.516i)8-s + (−0.309 − 0.951i)9-s + (−0.967 + 0.254i)12-s + (0.633 − 0.774i)13-s + (−0.974 − 0.226i)14-s + (0.198 + 0.980i)16-s + (−0.170 − 0.985i)17-s + (0.999 + 0.0285i)18-s + (−0.985 − 0.170i)19-s + (0.870 + 0.491i)21-s + ⋯
L(s)  = 1  + (−0.336 + 0.941i)2-s + (0.587 − 0.809i)3-s + (−0.774 − 0.633i)4-s + (0.564 + 0.825i)6-s + (0.113 + 0.993i)7-s + (0.856 − 0.516i)8-s + (−0.309 − 0.951i)9-s + (−0.967 + 0.254i)12-s + (0.633 − 0.774i)13-s + (−0.974 − 0.226i)14-s + (0.198 + 0.980i)16-s + (−0.170 − 0.985i)17-s + (0.999 + 0.0285i)18-s + (−0.985 − 0.170i)19-s + (0.870 + 0.491i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.847 - 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.847 - 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $0.847 - 0.530i$
Analytic conductor: \(14.0480\)
Root analytic conductor: \(14.0480\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3025,\ (0:\ ),\ 0.847 - 0.530i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.427852649 - 0.4100097373i\)
\(L(\frac12)\) \(\approx\) \(1.427852649 - 0.4100097373i\)
\(L(1)\) \(\approx\) \(1.026371925 + 0.07124733634i\)
\(L(1)\) \(\approx\) \(1.026371925 + 0.07124733634i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.336 + 0.941i)T \)
3 \( 1 + (0.587 - 0.809i)T \)
7 \( 1 + (0.113 + 0.993i)T \)
13 \( 1 + (0.633 - 0.774i)T \)
17 \( 1 + (-0.170 - 0.985i)T \)
19 \( 1 + (-0.985 - 0.170i)T \)
23 \( 1 + (-0.113 + 0.993i)T \)
29 \( 1 + (0.897 - 0.441i)T \)
31 \( 1 + (0.841 + 0.540i)T \)
37 \( 1 + (-0.931 + 0.362i)T \)
41 \( 1 + (0.959 - 0.281i)T \)
43 \( 1 + (0.755 + 0.654i)T \)
47 \( 1 + (0.791 - 0.610i)T \)
53 \( 1 + (0.113 + 0.993i)T \)
59 \( 1 + (-0.610 - 0.791i)T \)
61 \( 1 + (0.564 - 0.825i)T \)
67 \( 1 + (0.336 - 0.941i)T \)
71 \( 1 + (-0.142 - 0.989i)T \)
73 \( 1 + (-0.909 - 0.415i)T \)
79 \( 1 + (-0.921 + 0.389i)T \)
83 \( 1 + (0.676 + 0.736i)T \)
89 \( 1 + (0.985 - 0.170i)T \)
97 \( 1 + (0.856 - 0.516i)T \)
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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

−19.24556820689578882316833609219, −18.78188142459929858410609683955, −17.553863583765595615946455938872, −17.17718485678820525697165661772, −16.37564211788080069716356473798, −15.81596073299555148638106727710, −14.529121256727324962938591037110, −14.301509874147650337062128669617, −13.40991243637943747674053497140, −12.88521129620954639289729024551, −11.88579941971915418166233583083, −11.02767750158783290629428459162, −10.42513681388154362965053030882, −10.20392227068510233462030429986, −9.051482002229779080879141462344, −8.612683425824181777520456609246, −7.97539682357764837998067643300, −7.01612284570602078929477407156, −5.97131654379617071832134889240, −4.67420589906488922813400227761, −4.165299193791777919504227441167, −3.77371372061561882425822763225, −2.69497655387046699428633591555, −1.96930496638972630317083820266, −0.97684990434720063811836113604, 0.55752604292587168243465250718, 1.55074781381145278482303290674, 2.50595144528942216999626425551, 3.34113095388287018404706297619, 4.48803200469412260577379058358, 5.39273203670596584840113585713, 6.122044868715427274462719944368, 6.656147622577650054502474598876, 7.61539417484258386179383491322, 8.13151133245015639020299680701, 8.85465882656263868697858811415, 9.274446882947422860330913893501, 10.24310907329096326338463437322, 11.21039522940868757124524013183, 12.16246058250276982975736606130, 12.77064780579485780091114540278, 13.64058575834303805760190181135, 14.02084287470857260167327947417, 14.91452717977490400354883020539, 15.6468774017913047021417213719, 15.77874386914770609139871971000, 17.15166348870014322196073572399, 17.7215132733165724590718761925, 18.151773481768261420964460938811, 18.953772753620895247819807650573

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