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Properties

Label 1-2667-2667.677-r0-0-0
Degree $1$
Conductor $2667$
Sign $0.220 - 0.975i$
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  + (−0.826 − 0.563i)2-s + (0.365 + 0.930i)4-s + (−0.988 + 0.149i)5-s + (0.222 − 0.974i)8-s + (0.900 + 0.433i)10-s + (−0.270 + 0.962i)11-s + (0.0249 − 0.999i)13-s + (−0.733 + 0.680i)16-s + (0.980 + 0.198i)17-s − 19-s + (−0.5 − 0.866i)20-s + (0.766 − 0.642i)22-s + (−0.270 − 0.962i)23-s + (0.955 − 0.294i)25-s + (−0.583 + 0.811i)26-s + ⋯
L(s)  = 1  + (−0.826 − 0.563i)2-s + (0.365 + 0.930i)4-s + (−0.988 + 0.149i)5-s + (0.222 − 0.974i)8-s + (0.900 + 0.433i)10-s + (−0.270 + 0.962i)11-s + (0.0249 − 0.999i)13-s + (−0.733 + 0.680i)16-s + (0.980 + 0.198i)17-s − 19-s + (−0.5 − 0.866i)20-s + (0.766 − 0.642i)22-s + (−0.270 − 0.962i)23-s + (0.955 − 0.294i)25-s + (−0.583 + 0.811i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $0.220 - 0.975i$
Analytic conductor: \(12.3854\)
Root analytic conductor: \(12.3854\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (677, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2667,\ (0:\ ),\ 0.220 - 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4754253471 - 0.3797603925i\)
\(L(\frac12)\) \(\approx\) \(0.4754253471 - 0.3797603925i\)
\(L(1)\) \(\approx\) \(0.5647296077 - 0.1217491955i\)
\(L(1)\) \(\approx\) \(0.5647296077 - 0.1217491955i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
127 \( 1 \)
good2 \( 1 + (-0.826 - 0.563i)T \)
5 \( 1 + (-0.988 + 0.149i)T \)
11 \( 1 + (-0.270 + 0.962i)T \)
13 \( 1 + (0.0249 - 0.999i)T \)
17 \( 1 + (0.980 + 0.198i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.270 - 0.962i)T \)
29 \( 1 + (0.797 + 0.603i)T \)
31 \( 1 + (-0.980 + 0.198i)T \)
37 \( 1 + (0.766 - 0.642i)T \)
41 \( 1 + (-0.661 + 0.749i)T \)
43 \( 1 + (0.921 - 0.388i)T \)
47 \( 1 + (0.365 + 0.930i)T \)
53 \( 1 + (-0.995 + 0.0995i)T \)
59 \( 1 + (0.766 - 0.642i)T \)
61 \( 1 + (-0.955 - 0.294i)T \)
67 \( 1 + (-0.998 + 0.0498i)T \)
71 \( 1 + (-0.698 + 0.715i)T \)
73 \( 1 + (-0.0747 - 0.997i)T \)
79 \( 1 + (0.698 - 0.715i)T \)
83 \( 1 + (-0.411 + 0.911i)T \)
89 \( 1 + (-0.900 + 0.433i)T \)
97 \( 1 + (0.124 + 0.992i)T \)
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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

−19.36769666123205965580549550024, −18.752648220828558675399411331394, −18.25922180656926584569543455933, −17.051721778149876573842424719820, −16.71381543406896556214699566604, −15.99324201656267793069854783814, −15.473122487997638529358874155066, −14.638382535479215603164033408462, −14.02998759466142404472552836226, −13.14990251687574345254945567440, −11.98577889747374639973165214038, −11.53051125938500082782133560678, −10.8024701697937602919386362201, −10.03750640252225757051151701674, −9.08653660932573468057035754833, −8.56532326077772872831817051659, −7.80322772179458394787429364972, −7.27654004055813371642631897648, −6.33027964683310209507352670194, −5.65419969937849468172641498335, −4.693088499154217471218664693077, −3.84468759748730936743025851925, −2.8533090627210780782658012451, −1.71532402421198951728527899213, −0.7275729820161604977995547207, 0.38599352583950647830144032283, 1.476195042270245697231528078680, 2.55545756886257642772000922279, 3.212840273043804686036522322781, 4.10372990390983370142707606050, 4.80203990672555068204211001424, 6.09267990778275629908814329525, 7.02517970215465334137695348, 7.717719691336762083540931523652, 8.17348663282689821245805939673, 8.97772760527998666510698523716, 9.92655210262178713553673221978, 10.616760983681408602577814148270, 10.984362294759218691567962804, 12.15882222796073516577883315836, 12.46860034061920073291526255049, 13.004033996539439029454064758, 14.43610132076624528537032788688, 14.998050762189111607761092739621, 15.794334918431041723506439769476, 16.38649227391256078968012015309, 17.15098460411828077704344651660, 17.95986908383330872941316182632, 18.440017607057899587055889875634, 19.24932488408202819535636819623

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