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Properties

Label 2-867-51.23-c1-0-60
Degree $2$
Conductor $867$
Sign $-0.409 + 0.912i$
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  + (0.855 − 2.06i)2-s + (1.41 − 1.00i)3-s + (−2.12 − 2.12i)4-s + (2.35 + 1.57i)5-s + (−0.861 − 3.77i)6-s + (1.75 + 2.62i)7-s + (−2.06 + 0.855i)8-s + (0.992 − 2.83i)9-s + (5.25 − 3.51i)10-s + (−1.38 − 0.275i)11-s + (−5.12 − 0.871i)12-s + (−2.82 + 2.82i)13-s + (6.93 − 1.37i)14-s + (4.89 − 0.136i)15-s − 1.00i·16-s + ⋯
L(s)  = 1  + (0.605 − 1.46i)2-s + (0.815 − 0.578i)3-s + (−1.06 − 1.06i)4-s + (1.05 + 0.702i)5-s + (−0.351 − 1.54i)6-s + (0.664 + 0.993i)7-s + (−0.730 + 0.302i)8-s + (0.330 − 0.943i)9-s + (1.66 − 1.11i)10-s + (−0.418 − 0.0831i)11-s + (−1.47 − 0.251i)12-s + (−0.784 + 0.784i)13-s + (1.85 − 0.368i)14-s + (1.26 − 0.0352i)15-s − 0.250i·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $-0.409 + 0.912i$
Analytic conductor: \(6.92302\)
Root analytic conductor: \(2.63116\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (329, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :1/2),\ -0.409 + 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80431 - 2.78641i\)
\(L(\frac12)\) \(\approx\) \(1.80431 - 2.78641i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.41 + 1.00i)T \)
17 \( 1 \)
good2 \( 1 + (-0.855 + 2.06i)T + (-1.41 - 1.41i)T^{2} \)
5 \( 1 + (-2.35 - 1.57i)T + (1.91 + 4.61i)T^{2} \)
7 \( 1 + (-1.75 - 2.62i)T + (-2.67 + 6.46i)T^{2} \)
11 \( 1 + (1.38 + 0.275i)T + (10.1 + 4.20i)T^{2} \)
13 \( 1 + (2.82 - 2.82i)T - 13iT^{2} \)
19 \( 1 + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-1.37 + 6.93i)T + (-21.2 - 8.80i)T^{2} \)
29 \( 1 + (-1.57 + 2.35i)T + (-11.0 - 26.7i)T^{2} \)
31 \( 1 + (-0.616 - 3.10i)T + (-28.6 + 11.8i)T^{2} \)
37 \( 1 + (6.20 - 1.23i)T + (34.1 - 14.1i)T^{2} \)
41 \( 1 + (-4.70 + 3.14i)T + (15.6 - 37.8i)T^{2} \)
43 \( 1 + (3.69 - 1.53i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + (-6.32 - 6.32i)T + 47iT^{2} \)
53 \( 1 + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (8.26 - 3.42i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (5.25 - 3.51i)T + (23.3 - 56.3i)T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 + (-2.48 - 12.4i)T + (-65.5 + 27.1i)T^{2} \)
73 \( 1 + (-27.9 - 67.4i)T^{2} \)
79 \( 1 + (0.616 - 3.10i)T + (-72.9 - 30.2i)T^{2} \)
83 \( 1 + (8.26 + 3.42i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (9.48 - 9.48i)T - 89iT^{2} \)
97 \( 1 + (-10.5 - 7.02i)T + (37.1 + 89.6i)T^{2} \)
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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

−10.06419426320458970754817989722, −9.251970576384310569353625337674, −8.537323096659280443380451807810, −7.29644253274635753408886351608, −6.32126889621186172530024537250, −5.25069214818755322255705920944, −4.23800373184242930053596807700, −2.65809883053530275723345614061, −2.56577414878775105383658321382, −1.56730484608503064989588553860, 1.78025055212393433175717814244, 3.40977264051029411263019842771, 4.61858687252583065453872474822, 5.07788792071054961635218101049, 5.83861646396282583696192173413, 7.27770714549655405718205855963, 7.66561110960589121262921263488, 8.524759629687437002473441220770, 9.396635212508068285594057961705, 10.17149863646356647542224016208

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