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Properties

Label 1-2668-2668.163-r0-0-0
Degree $1$
Conductor $2668$
Sign $0.0700 - 0.997i$
Analytic cond. $12.3901$
Root an. cond. $12.3901$
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  + (0.872 − 0.488i)3-s + (−0.101 − 0.994i)5-s + (0.933 − 0.359i)7-s + (0.523 − 0.852i)9-s + (0.639 + 0.768i)11-s + (−0.557 + 0.830i)13-s + (−0.574 − 0.818i)15-s + (0.755 − 0.654i)17-s + (−0.806 + 0.591i)19-s + (0.639 − 0.768i)21-s + (−0.979 + 0.202i)25-s + (0.0407 − 0.999i)27-s + (−0.925 + 0.377i)31-s + (0.933 + 0.359i)33-s + (−0.452 − 0.891i)35-s + ⋯
L(s)  = 1  + (0.872 − 0.488i)3-s + (−0.101 − 0.994i)5-s + (0.933 − 0.359i)7-s + (0.523 − 0.852i)9-s + (0.639 + 0.768i)11-s + (−0.557 + 0.830i)13-s + (−0.574 − 0.818i)15-s + (0.755 − 0.654i)17-s + (−0.806 + 0.591i)19-s + (0.639 − 0.768i)21-s + (−0.979 + 0.202i)25-s + (0.0407 − 0.999i)27-s + (−0.925 + 0.377i)31-s + (0.933 + 0.359i)33-s + (−0.452 − 0.891i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2668 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0700 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2668 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0700 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2668\)    =    \(2^{2} \cdot 23 \cdot 29\)
Sign: $0.0700 - 0.997i$
Analytic conductor: \(12.3901\)
Root analytic conductor: \(12.3901\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2668} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2668,\ (0:\ ),\ 0.0700 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.967456233 - 1.834078899i\)
\(L(\frac12)\) \(\approx\) \(1.967456233 - 1.834078899i\)
\(L(1)\) \(\approx\) \(1.483587778 - 0.6048796268i\)
\(L(1)\) \(\approx\) \(1.483587778 - 0.6048796268i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good3 \( 1 + (0.872 - 0.488i)T \)
5 \( 1 + (-0.101 - 0.994i)T \)
7 \( 1 + (0.933 - 0.359i)T \)
11 \( 1 + (0.639 + 0.768i)T \)
13 \( 1 + (-0.557 + 0.830i)T \)
17 \( 1 + (0.755 - 0.654i)T \)
19 \( 1 + (-0.806 + 0.591i)T \)
31 \( 1 + (-0.925 + 0.377i)T \)
37 \( 1 + (0.852 + 0.523i)T \)
41 \( 1 + (0.540 - 0.841i)T \)
43 \( 1 + (0.925 + 0.377i)T \)
47 \( 1 + (0.433 - 0.900i)T \)
53 \( 1 + (0.685 - 0.728i)T \)
59 \( 1 + (0.142 - 0.989i)T \)
61 \( 1 + (-0.998 - 0.0611i)T \)
67 \( 1 + (-0.768 - 0.639i)T \)
71 \( 1 + (0.0203 - 0.999i)T \)
73 \( 1 + (0.983 - 0.182i)T \)
79 \( 1 + (-0.953 + 0.301i)T \)
83 \( 1 + (-0.339 - 0.940i)T \)
89 \( 1 + (0.574 - 0.818i)T \)
97 \( 1 + (0.699 + 0.714i)T \)
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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

−19.5492565105598962795129775765, −18.85429600730965417536252187755, −18.23086329082097643014420313185, −17.36805983756921698781647417944, −16.681405854113668713538237359161, −15.66574971263927464362907464755, −15.016758966155377506284109914520, −14.58671675241169878647259875765, −14.16014383771778494749767370461, −13.20701310061826839440689336725, −12.37028230641185032134499445922, −11.3348372148305174154561367993, −10.828402793240055416896736296500, −10.21433079934582734211061156371, −9.25532384785810246842775524796, −8.62670012976159810114591683126, −7.746514025638522915901417353, −7.43007871104236728301618773685, −6.13844527845458143540626905055, −5.51422255210221992422365467876, −4.36959821709114370580629311997, −3.79644494958921484006109896842, −2.80444374716559327010703604267, −2.367870260919513465421442487923, −1.21048130238185090633759343925, 0.795075739861530515658859395049, 1.74729982777297331364973618531, 2.10309567331006984805535077204, 3.51729877882777123307781327644, 4.304631065412768665027170092841, 4.77849090167898495339928236260, 5.87749562162345146269032403335, 6.99018368504292689398005808990, 7.537300121400677777389098947457, 8.185151220928308862837824575013, 9.02957459426812591302370412919, 9.44903500179626247977059961161, 10.34577102310010040087174995297, 11.5488149963933701201125049404, 12.13904956106908770013952390265, 12.638504217173421460285159824767, 13.52380938394371678835669974243, 14.33631583573554143545795678973, 14.58995187100524711747726211220, 15.46274140459363390489673252347, 16.513634108143738818010628672937, 16.9914567010704655901435920304, 17.74211161236944973621610335138, 18.50082763295840384763473992508, 19.27540743994829349260357981320

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