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Properties

Label 1-1312-1312.1125-r0-0-0
Degree $1$
Conductor $1312$
Sign $0.882 - 0.471i$
Analytic cond. $6.09290$
Root an. cond. $6.09290$
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  + (−0.707 + 0.707i)3-s + (0.891 − 0.453i)5-s + (0.587 − 0.809i)7-s − i·9-s + (0.453 − 0.891i)11-s + (−0.156 + 0.987i)13-s + (−0.309 + 0.951i)15-s + (−0.309 − 0.951i)17-s + (−0.987 + 0.156i)19-s + (0.156 + 0.987i)21-s + (0.587 + 0.809i)23-s + (0.587 − 0.809i)25-s + (0.707 + 0.707i)27-s + (0.453 + 0.891i)29-s + (0.309 + 0.951i)31-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s + (0.891 − 0.453i)5-s + (0.587 − 0.809i)7-s − i·9-s + (0.453 − 0.891i)11-s + (−0.156 + 0.987i)13-s + (−0.309 + 0.951i)15-s + (−0.309 − 0.951i)17-s + (−0.987 + 0.156i)19-s + (0.156 + 0.987i)21-s + (0.587 + 0.809i)23-s + (0.587 − 0.809i)25-s + (0.707 + 0.707i)27-s + (0.453 + 0.891i)29-s + (0.309 + 0.951i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1312 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1312 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1312\)    =    \(2^{5} \cdot 41\)
Sign: $0.882 - 0.471i$
Analytic conductor: \(6.09290\)
Root analytic conductor: \(6.09290\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1312} (1125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1312,\ (0:\ ),\ 0.882 - 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.505406151 - 0.3767562063i\)
\(L(\frac12)\) \(\approx\) \(1.505406151 - 0.3767562063i\)
\(L(1)\) \(\approx\) \(1.096880103 - 0.04785512921i\)
\(L(1)\) \(\approx\) \(1.096880103 - 0.04785512921i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
41 \( 1 \)
good3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (0.891 - 0.453i)T \)
7 \( 1 + (0.587 - 0.809i)T \)
11 \( 1 + (0.453 - 0.891i)T \)
13 \( 1 + (-0.156 + 0.987i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (-0.987 + 0.156i)T \)
23 \( 1 + (0.587 + 0.809i)T \)
29 \( 1 + (0.453 + 0.891i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (0.891 - 0.453i)T \)
43 \( 1 + (0.987 + 0.156i)T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (-0.891 + 0.453i)T \)
59 \( 1 + (-0.987 - 0.156i)T \)
61 \( 1 + (0.987 - 0.156i)T \)
67 \( 1 + (0.453 + 0.891i)T \)
71 \( 1 + (-0.951 + 0.309i)T \)
73 \( 1 - iT \)
79 \( 1 - T \)
83 \( 1 + (0.707 - 0.707i)T \)
89 \( 1 + (0.587 - 0.809i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

−21.150859720000194949890231817997, −20.308566096034690503985289236739, −19.174707157554040683941154623670, −18.708052395368081680050143074796, −17.726796095345152165950815580682, −17.46902585905335859855117788954, −16.888071004738610844975260607052, −15.439716156521282394932592532177, −14.92232591736614331267403050429, −14.144275196761020467147137495984, −12.958172628856404548342293117163, −12.73956158865567755385850814974, −11.76552496647584259448181380508, −10.88669333744889969109169415331, −10.32232667760317373660936289024, −9.31259818766578913029871484615, −8.300346259987605631558138326619, −7.55314246094647292596966354617, −6.365307565446484269170355333660, −6.145249017997810043020263073548, −5.127108049548574189591327084275, −4.33246055799056353610116227791, −2.53899607163820029357027206149, −2.19723791998184791584398885658, −1.10348913254888789507920593367, 0.78605015874928179201157607370, 1.67120533830544353469747155130, 3.07002681244401440779323110730, 4.24691503057133903156369345172, 4.739711588882063665951288164712, 5.6442494608153387216032493691, 6.46707662138184815334391435065, 7.22124672184406480677867065794, 8.701525492869662776932861720707, 9.132102523273626781183725197468, 10.03439828146032956247971922622, 10.85979004986756927725731293410, 11.37904437929482501288068094259, 12.28853380235842458668199767302, 13.29033067772182303481938627499, 14.1386651470461989492848420206, 14.52518107432540859394306585544, 15.87291869077254482659511467714, 16.43988588435984767258320414269, 17.119593335234790790036585897808, 17.53881117687767170533387356612, 18.425815077099932609531178904610, 19.47416485363443062681689898874, 20.39209167618956651886125351998, 21.07771835788394755074394065622

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