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Properties

Label 1-2667-2667.86-r0-0-0
Degree $1$
Conductor $2667$
Sign $-0.821 - 0.569i$
Analytic cond. $12.3854$
Root an. cond. $12.3854$
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  + (0.733 − 0.680i)2-s + (0.0747 − 0.997i)4-s + (0.826 − 0.563i)5-s + (−0.623 − 0.781i)8-s + (0.222 − 0.974i)10-s + (0.998 + 0.0498i)11-s + (−0.411 − 0.911i)13-s + (−0.988 − 0.149i)16-s + (0.969 − 0.246i)17-s + 19-s + (−0.5 − 0.866i)20-s + (0.766 − 0.642i)22-s + (−0.998 + 0.0498i)23-s + (0.365 − 0.930i)25-s + (−0.921 − 0.388i)26-s + ⋯
L(s)  = 1  + (0.733 − 0.680i)2-s + (0.0747 − 0.997i)4-s + (0.826 − 0.563i)5-s + (−0.623 − 0.781i)8-s + (0.222 − 0.974i)10-s + (0.998 + 0.0498i)11-s + (−0.411 − 0.911i)13-s + (−0.988 − 0.149i)16-s + (0.969 − 0.246i)17-s + 19-s + (−0.5 − 0.866i)20-s + (0.766 − 0.642i)22-s + (−0.998 + 0.0498i)23-s + (0.365 − 0.930i)25-s + (−0.921 − 0.388i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.821 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.821 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $-0.821 - 0.569i$
Analytic conductor: \(12.3854\)
Root analytic conductor: \(12.3854\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2667,\ (0:\ ),\ -0.821 - 0.569i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9246161292 - 2.957793747i\)
\(L(\frac12)\) \(\approx\) \(0.9246161292 - 2.957793747i\)
\(L(1)\) \(\approx\) \(1.406776431 - 1.199587140i\)
\(L(1)\) \(\approx\) \(1.406776431 - 1.199587140i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
127 \( 1 \)
good2 \( 1 + (0.733 - 0.680i)T \)
5 \( 1 + (0.826 - 0.563i)T \)
11 \( 1 + (0.998 + 0.0498i)T \)
13 \( 1 + (-0.411 - 0.911i)T \)
17 \( 1 + (0.969 - 0.246i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.998 + 0.0498i)T \)
29 \( 1 + (-0.0249 - 0.999i)T \)
31 \( 1 + (-0.969 - 0.246i)T \)
37 \( 1 + (0.766 - 0.642i)T \)
41 \( 1 + (-0.270 + 0.962i)T \)
43 \( 1 + (-0.878 - 0.478i)T \)
47 \( 1 + (-0.0747 + 0.997i)T \)
53 \( 1 + (-0.124 + 0.992i)T \)
59 \( 1 + (0.766 - 0.642i)T \)
61 \( 1 + (0.365 + 0.930i)T \)
67 \( 1 + (0.661 + 0.749i)T \)
71 \( 1 + (-0.542 - 0.840i)T \)
73 \( 1 + (0.955 - 0.294i)T \)
79 \( 1 + (0.542 + 0.840i)T \)
83 \( 1 + (-0.797 - 0.603i)T \)
89 \( 1 + (-0.222 - 0.974i)T \)
97 \( 1 + (0.853 + 0.521i)T \)
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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

−19.72477725146585379344017033828, −18.63021847038276296065871570741, −18.12024470880323980864948382225, −17.30519545982427915091737984466, −16.62955283605117036735141526888, −16.25166511136130014648058353652, −15.10228915202315802647830164514, −14.37561278205479633673158046749, −14.217492463579677801807372895617, −13.43846936655846986711613490082, −12.53344130030032795878832485760, −11.83730944001569246234163417604, −11.25889520849437700419049190500, −10.07700904387764559809124247266, −9.46986846304527993306348581123, −8.68067224099412715949683939931, −7.69964184947154311685905836819, −6.884539785656305694089881668695, −6.49064651007472891308856645879, −5.55346130977974735559615918500, −5.03236051764585674472379096797, −3.81965722959584688503815813566, −3.38368483904237958904736699118, −2.26897132679529377419872210273, −1.4777870249508115692490199383, 0.74506806651008358376975718515, 1.489135955996207791888066360760, 2.372884502785875916314541057507, 3.24511099710097606804746763800, 4.07640615527687946311281372329, 4.938163476452066324259211312289, 5.69987444715572607574599393031, 6.08641078477551658989333697440, 7.215687855169849399305628074511, 8.14717828926954839673804779156, 9.29607540794661121002057174280, 9.73289995387089879486777503570, 10.25200631000736607851111256751, 11.362000965168697352166231217310, 11.96349511900907085204760183879, 12.61506083057229046778194171057, 13.25826831349852102823163465906, 14.06773924304322018212493425474, 14.44920960900267828295856876480, 15.30494613511837303663226110807, 16.227838120258989920403235955624, 16.857554508131284276485188760892, 17.79747411943375266627682689016, 18.28983616190881443883002588411, 19.24307456054857700924864086159

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