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Properties

Label 1-3131-3131.1657-r1-0-0
Degree $1$
Conductor $3131$
Sign $0.264 + 0.964i$
Analytic cond. $336.472$
Root an. cond. $336.472$
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  + (0.951 + 0.309i)2-s + (0.207 − 0.978i)3-s + (0.809 + 0.587i)4-s + (−0.978 + 0.207i)5-s + (0.5 − 0.866i)6-s + (−0.866 − 0.5i)7-s + (0.587 + 0.809i)8-s + (−0.913 − 0.406i)9-s + (−0.994 − 0.104i)10-s + (−0.207 − 0.978i)11-s + (0.743 − 0.669i)12-s + (0.104 − 0.994i)13-s + (−0.669 − 0.743i)14-s + i·15-s + (0.309 + 0.951i)16-s + (−0.669 − 0.743i)17-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)2-s + (0.207 − 0.978i)3-s + (0.809 + 0.587i)4-s + (−0.978 + 0.207i)5-s + (0.5 − 0.866i)6-s + (−0.866 − 0.5i)7-s + (0.587 + 0.809i)8-s + (−0.913 − 0.406i)9-s + (−0.994 − 0.104i)10-s + (−0.207 − 0.978i)11-s + (0.743 − 0.669i)12-s + (0.104 − 0.994i)13-s + (−0.669 − 0.743i)14-s + i·15-s + (0.309 + 0.951i)16-s + (−0.669 − 0.743i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3131 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.264 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3131 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.264 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3131\)    =    \(31 \cdot 101\)
Sign: $0.264 + 0.964i$
Analytic conductor: \(336.472\)
Root analytic conductor: \(336.472\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3131} (1657, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3131,\ (1:\ ),\ 0.264 + 0.964i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1554795042 + 0.1186017608i\)
\(L(\frac12)\) \(\approx\) \(0.1554795042 + 0.1186017608i\)
\(L(1)\) \(\approx\) \(1.142032024 - 0.3531406419i\)
\(L(1)\) \(\approx\) \(1.142032024 - 0.3531406419i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
101 \( 1 \)
good2 \( 1 + (0.951 + 0.309i)T \)
3 \( 1 + (0.207 - 0.978i)T \)
5 \( 1 + (-0.978 + 0.207i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (-0.207 - 0.978i)T \)
13 \( 1 + (0.104 - 0.994i)T \)
17 \( 1 + (-0.669 - 0.743i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.951 - 0.309i)T \)
37 \( 1 + (-0.978 - 0.207i)T \)
41 \( 1 + (-0.994 - 0.104i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 - T \)
53 \( 1 + (0.406 - 0.913i)T \)
59 \( 1 + (0.207 - 0.978i)T \)
61 \( 1 + (-0.587 - 0.809i)T \)
67 \( 1 + (0.743 + 0.669i)T \)
71 \( 1 + (-0.104 + 0.994i)T \)
73 \( 1 + (-0.406 + 0.913i)T \)
79 \( 1 + (0.913 - 0.406i)T \)
83 \( 1 + (-0.406 + 0.913i)T \)
89 \( 1 + (-0.587 - 0.809i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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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

−19.03526856903547635942029054071, −18.092614868508483974979389007497, −16.82658576291031070700761921009, −16.26518710425654232795267103085, −15.67604079450452432163343152046, −15.21175796112586927700465821431, −14.7115715504242602067168073260, −13.709791519327286682103959580957, −13.08326281256398772691881916465, −12.1459116292848931465110553979, −11.81038081383303278229427808623, −10.9703508963684189150468915713, −10.28541576936373835598756692420, −9.4424514165707241323258123801, −8.93867310519057097024689013803, −7.83801273665777718495563085752, −6.965407272279336723968285041699, −6.242514594923037304459497330933, −5.25843353364209770228432987276, −4.60311241037817410070793429069, −3.982469302372847003429592397887, −3.389395355207954110361985064006, −2.54507015326304093634647159076, −1.683396220664117258326807752411, −0.028843193313044669846347780869, 0.60835975567793490182728156905, 1.93824874428326086362507275281, 3.04246612698326647177466626872, 3.35693961461243792525149466433, 4.05281390763471510010573029579, 5.38478315354298110359888692028, 5.879837925955729649789954870971, 6.90396080966145781788533341069, 7.15005770118020995283732248748, 8.165739510421207911303701711105, 8.35691999197560384653474775077, 9.77779060504088943307941778880, 10.843128063274779500268327336236, 11.39315801450475455463744009023, 12.144565750711183999679333568944, 12.699245765517256057323297441081, 13.44060028418691782940704648447, 13.87589732256131350837443993587, 14.59868719333706041748052991213, 15.56220933604134763452596459289, 15.96584837609489765446694223115, 16.64464199682365876895670430193, 17.49663244776056922813172492555, 18.41242492210882759511805879356, 19.01939012460319768740645628468

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