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Properties

Label 2-867-17.8-c1-0-23
Degree $2$
Conductor $867$
Sign $0.940 + 0.339i$
Analytic cond. $6.92302$
Root an. cond. $2.63116$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s + (0.382 − 0.923i)3-s + 0.999i·4-s + (−0.382 − 0.923i)6-s + (3.69 − 1.53i)7-s + (2.12 + 2.12i)8-s + (−0.707 − 0.707i)9-s + (1.53 + 3.69i)11-s + (0.923 + 0.382i)12-s + 2i·13-s + (1.53 − 3.69i)14-s + 1.00·16-s − 18-s + (−2.82 + 2.82i)19-s − 4i·21-s + (3.69 + 1.53i)22-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.220 − 0.533i)3-s + 0.499i·4-s + (−0.156 − 0.377i)6-s + (1.39 − 0.578i)7-s + (0.750 + 0.750i)8-s + (−0.235 − 0.235i)9-s + (0.461 + 1.11i)11-s + (0.266 + 0.110i)12-s + 0.554i·13-s + (0.409 − 0.987i)14-s + 0.250·16-s − 0.235·18-s + (−0.648 + 0.648i)19-s − 0.872i·21-s + (0.787 + 0.326i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $0.940 + 0.339i$
Analytic conductor: \(6.92302\)
Root analytic conductor: \(2.63116\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (688, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :1/2),\ 0.940 + 0.339i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.56702 - 0.448802i\)
\(L(\frac12)\) \(\approx\) \(2.56702 - 0.448802i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.382 + 0.923i)T \)
17 \( 1 \)
good2 \( 1 + (-0.707 + 0.707i)T - 2iT^{2} \)
5 \( 1 + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (-3.69 + 1.53i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-1.53 - 3.69i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
19 \( 1 + (2.82 - 2.82i)T - 19iT^{2} \)
23 \( 1 + (-1.53 - 3.69i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-1.53 + 3.69i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (-3.06 + 7.39i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-7.39 + 3.06i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (2.82 + 2.82i)T + 43iT^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 + (-4.24 + 4.24i)T - 53iT^{2} \)
59 \( 1 + (-8.48 - 8.48i)T + 59iT^{2} \)
61 \( 1 + (7.39 - 3.06i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 + (-4.59 + 11.0i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (1.53 + 3.69i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (8.48 - 8.48i)T - 83iT^{2} \)
89 \( 1 - 10iT - 89T^{2} \)
97 \( 1 + (14.7 + 6.12i)T + (68.5 + 68.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.34599845751109097781714026072, −9.184661666814427839695692712381, −8.214371619454900060117605815918, −7.60234804140314003402562118467, −6.95318737065085679869268284230, −5.54075207406549870127039516630, −4.31989187592904266019259251662, −4.00534454798458534180872564725, −2.30807065259819462227436892699, −1.62898632020941700949707464929, 1.28800883655372998106192910866, 2.80691699989578269407417718801, 4.21577916264664443275224538689, 4.92618044183224222627875115986, 5.70226183376600136859178903830, 6.48933124562573300765523502034, 7.76451161433833617753417975614, 8.498024613265010138643897232446, 9.242545249902496706633722129741, 10.29597752262362615685963042928

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