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Properties

Label 1-277-277.6-r1-0-0
Degree $1$
Conductor $277$
Sign $0.694 - 0.719i$
Analytic cond. $29.7677$
Root an. cond. $29.7677$
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  + (−0.887 + 0.460i)2-s + (0.377 + 0.926i)3-s + (0.576 − 0.816i)4-s + (0.225 + 0.974i)5-s + (−0.761 − 0.648i)6-s + (−0.291 − 0.956i)7-s + (−0.136 + 0.990i)8-s + (−0.715 + 0.699i)9-s + (−0.648 − 0.761i)10-s + (−0.999 + 0.0227i)11-s + (0.974 + 0.225i)12-s + (−0.962 + 0.269i)13-s + (0.699 + 0.715i)14-s + (−0.816 + 0.576i)15-s + (−0.334 − 0.942i)16-s + (0.993 + 0.113i)17-s + ⋯
L(s)  = 1  + (−0.887 + 0.460i)2-s + (0.377 + 0.926i)3-s + (0.576 − 0.816i)4-s + (0.225 + 0.974i)5-s + (−0.761 − 0.648i)6-s + (−0.291 − 0.956i)7-s + (−0.136 + 0.990i)8-s + (−0.715 + 0.699i)9-s + (−0.648 − 0.761i)10-s + (−0.999 + 0.0227i)11-s + (0.974 + 0.225i)12-s + (−0.962 + 0.269i)13-s + (0.699 + 0.715i)14-s + (−0.816 + 0.576i)15-s + (−0.334 − 0.942i)16-s + (0.993 + 0.113i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(277\)
Sign: $0.694 - 0.719i$
Analytic conductor: \(29.7677\)
Root analytic conductor: \(29.7677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{277} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 277,\ (1:\ ),\ 0.694 - 0.719i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3925635723 - 0.1667746858i\)
\(L(\frac12)\) \(\approx\) \(0.3925635723 - 0.1667746858i\)
\(L(1)\) \(\approx\) \(0.5769196755 + 0.2794383474i\)
\(L(1)\) \(\approx\) \(0.5769196755 + 0.2794383474i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad277 \( 1 \)
good2 \( 1 + (-0.887 + 0.460i)T \)
3 \( 1 + (0.377 + 0.926i)T \)
5 \( 1 + (0.225 + 0.974i)T \)
7 \( 1 + (-0.291 - 0.956i)T \)
11 \( 1 + (-0.999 + 0.0227i)T \)
13 \( 1 + (-0.962 + 0.269i)T \)
17 \( 1 + (0.993 + 0.113i)T \)
19 \( 1 + (0.682 - 0.730i)T \)
23 \( 1 + (-0.974 + 0.225i)T \)
29 \( 1 + (0.419 - 0.907i)T \)
31 \( 1 + (0.956 + 0.291i)T \)
37 \( 1 + (-0.816 - 0.576i)T \)
41 \( 1 + (0.917 - 0.398i)T \)
43 \( 1 + (0.0455 - 0.998i)T \)
47 \( 1 + (-0.803 - 0.595i)T \)
53 \( 1 + (-0.439 + 0.898i)T \)
59 \( 1 + (-0.854 - 0.519i)T \)
61 \( 1 + (-0.519 + 0.854i)T \)
67 \( 1 + (0.983 + 0.181i)T \)
71 \( 1 + (0.613 - 0.789i)T \)
73 \( 1 + (-0.942 - 0.334i)T \)
79 \( 1 + (0.291 - 0.956i)T \)
83 \( 1 + (0.648 - 0.761i)T \)
89 \( 1 + (0.949 - 0.313i)T \)
97 \( 1 + (0.926 - 0.377i)T \)
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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

−25.56954671261657105621070996545, −24.68914634084094391616643089566, −24.283398023880227174220185675097, −22.84525037193676852417157626680, −21.53168425260807075304687951236, −20.76603699380160145648386934160, −19.947596479121033999260202379136, −19.0985261615880687028106596460, −18.29895435446415216621888828823, −17.60731738229973873337549534928, −16.48969055612045074598560662063, −15.67170447795372140196255150038, −14.2556261838938884161491906652, −12.921896058666706007494602680336, −12.39499514609141104816811641980, −11.78002301390332503470480458946, −10.03409160746009683784474941503, −9.37433331738681551177471572030, −8.135047738927118045709576908215, −7.86561388900243131874129752186, −6.307162822371797734052611848993, −5.1936071440208119363204480068, −3.16988130598717931814756675930, −2.29041411974500833010016852699, −1.14898452158048943871314467721, 0.17452901175271344982346228047, 2.271478754434344836487934265534, 3.28306917598377777157111853167, 4.83053141763900975624522681689, 5.99434288860915069969866600532, 7.32921414772694543164548388394, 7.863228564181088339792075429110, 9.38081898921905971923204468111, 10.20145262414173096322425703821, 10.49108956713817306700958258883, 11.73516960670419044597647071172, 13.78353055161139168958851753680, 14.26106370992778855669811548992, 15.37540463569067533080835281758, 16.00294234870301062502183066075, 17.06370129342449072693758920887, 17.78728334965235973017785964916, 19.03893946255882721826018068603, 19.65328484163505795418647585127, 20.655524996227102907892615563573, 21.55248607437791395545888237327, 22.71169632062623154776255814851, 23.474347158499951274544688596722, 24.68022829820780097818152300663, 25.83556614042178184801215826043

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