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Properties

Label 1-2015-2015.1048-r0-0-0
Degree $1$
Conductor $2015$
Sign $0.0770 - 0.997i$
Analytic cond. $9.35762$
Root an. cond. $9.35762$
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  + 2-s + (−0.866 − 0.5i)3-s + 4-s + (−0.866 − 0.5i)6-s + (0.5 − 0.866i)7-s + 8-s + (0.5 + 0.866i)9-s + (−0.866 + 0.5i)11-s + (−0.866 − 0.5i)12-s + (0.5 − 0.866i)14-s + 16-s + (−0.866 − 0.5i)17-s + (0.5 + 0.866i)18-s + (0.866 + 0.5i)19-s + (−0.866 + 0.5i)21-s + (−0.866 + 0.5i)22-s + ⋯
L(s)  = 1  + 2-s + (−0.866 − 0.5i)3-s + 4-s + (−0.866 − 0.5i)6-s + (0.5 − 0.866i)7-s + 8-s + (0.5 + 0.866i)9-s + (−0.866 + 0.5i)11-s + (−0.866 − 0.5i)12-s + (0.5 − 0.866i)14-s + 16-s + (−0.866 − 0.5i)17-s + (0.5 + 0.866i)18-s + (0.866 + 0.5i)19-s + (−0.866 + 0.5i)21-s + (−0.866 + 0.5i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0770 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0770 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $0.0770 - 0.997i$
Analytic conductor: \(9.35762\)
Root analytic conductor: \(9.35762\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (1048, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (0:\ ),\ 0.0770 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.698401680 - 1.572179787i\)
\(L(\frac12)\) \(\approx\) \(1.698401680 - 1.572179787i\)
\(L(1)\) \(\approx\) \(1.477428388 - 0.4791396299i\)
\(L(1)\) \(\approx\) \(1.477428388 - 0.4791396299i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
31 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 - iT \)
29 \( 1 - T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.866 - 0.5i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 - T \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.866 - 0.5i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + iT \)
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

−20.46082816813427979491224213833, −19.51352356782767689111388176540, −18.53284697358010556562560668436, −17.86432123982778681240863233818, −17.08935126734455602427533248675, −16.19703248759522065206093642200, −15.58359321531824615294572906260, −15.224258591045071959582614133769, −14.352232506343820760305879911016, −13.29358000539952386760812775904, −12.851719141396663172164158632797, −11.833426127822688025011041521930, −11.36386211347694121746790045847, −10.859977863180522997426792488826, −9.89999676083085431204027697074, −8.98337231559458047839361439755, −7.90696003478567645770315530007, −7.083848664817391529228398254384, −6.03668770348610064782364313292, −5.5817696691648988850240361728, −4.95186103141605258072444079842, −4.1548333423285931863203847895, −3.18873288819056094449168110642, −2.32253815743235564556975012753, −1.20593159534030608114831721098, 0.64312409621285073422549623197, 1.79931470369906844612232482192, 2.49383869402213073321490628107, 3.78891381783939402970046690931, 4.629894730920573839669998636205, 5.13339691247639113932552031557, 5.99356208413109483523439678161, 6.84724163219390930049314668456, 7.50658829478958628979274288515, 7.97415072832630657543968555934, 9.58776916693794136075701519702, 10.61255032389020024935948179290, 10.93459921682129967707511078454, 11.705643039327496573008916613306, 12.53837990909213361858742722176, 13.09418164884441662310866273331, 13.75931449797904026416801141219, 14.475741186903429609822533911299, 15.36379629242715003944342960625, 16.31979715584919666114783042109, 16.52927969515110398644757047865, 17.721218556799333172674693562376, 18.04093968138668067695370335421, 19.08901183591260446840488910480, 19.99029559411657695624697018927

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