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Properties

Label 2-867-51.23-c1-0-59
Degree $2$
Conductor $867$
Sign $0.844 + 0.534i$
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  + (1.69 − 0.337i)3-s + (1.41 + 1.41i)4-s + (−2.90 − 4.34i)7-s + (2.77 − 1.14i)9-s + (2.88 + 1.92i)12-s + (3.00 − 3.00i)13-s + 4.00i·16-s + (4.43 + 1.83i)19-s + (−6.39 − 6.39i)21-s + (−1.91 − 4.61i)25-s + (4.32 − 2.88i)27-s + (2.03 − 10.2i)28-s + (0.416 + 2.09i)31-s + (5.54 + 2.29i)36-s + (3.03 − 0.602i)37-s + ⋯
L(s)  = 1  + (0.980 − 0.195i)3-s + (0.707 + 0.707i)4-s + (−1.09 − 1.64i)7-s + (0.923 − 0.382i)9-s + (0.831 + 0.555i)12-s + (0.833 − 0.833i)13-s + 1.00i·16-s + (1.01 + 0.421i)19-s + (−1.39 − 1.39i)21-s + (−0.382 − 0.923i)25-s + (0.831 − 0.555i)27-s + (0.385 − 1.93i)28-s + (0.0747 + 0.376i)31-s + (0.923 + 0.382i)36-s + (0.498 − 0.0991i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.844 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $0.844 + 0.534i$
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: \((2,\ 867,\ (\ :1/2),\ 0.844 + 0.534i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.32772 - 0.674873i\)
\(L(\frac12)\) \(\approx\) \(2.32772 - 0.674873i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.69 + 0.337i)T \)
17 \( 1 \)
good2 \( 1 + (-1.41 - 1.41i)T^{2} \)
5 \( 1 + (1.91 + 4.61i)T^{2} \)
7 \( 1 + (2.90 + 4.34i)T + (-2.67 + 6.46i)T^{2} \)
11 \( 1 + (10.1 + 4.20i)T^{2} \)
13 \( 1 + (-3.00 + 3.00i)T - 13iT^{2} \)
19 \( 1 + (-4.43 - 1.83i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-21.2 - 8.80i)T^{2} \)
29 \( 1 + (-11.0 - 26.7i)T^{2} \)
31 \( 1 + (-0.416 - 2.09i)T + (-28.6 + 11.8i)T^{2} \)
37 \( 1 + (-3.03 + 0.602i)T + (34.1 - 14.1i)T^{2} \)
41 \( 1 + (15.6 - 37.8i)T^{2} \)
43 \( 1 + (0.509 - 0.211i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (10.4 - 6.99i)T + (23.3 - 56.3i)T^{2} \)
67 \( 1 - 13.8iT - 67T^{2} \)
71 \( 1 + (-65.5 + 27.1i)T^{2} \)
73 \( 1 + (-0.962 + 1.44i)T + (-27.9 - 67.4i)T^{2} \)
79 \( 1 + (-1.01 + 5.09i)T + (-72.9 - 30.2i)T^{2} \)
83 \( 1 + (58.6 + 58.6i)T^{2} \)
89 \( 1 - 89iT^{2} \)
97 \( 1 + (-14.8 - 9.89i)T + (37.1 + 89.6i)T^{2} \)
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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

−10.20956830478033036200160375996, −9.230767525380282303369895941493, −8.115930350266711037245340648388, −7.60321368908738632326070821192, −6.87711517601589760270482647952, −6.09844454863020418332914465453, −4.17961860060580012847989305729, −3.51000868392703878839270659768, −2.82722479860047747466049265401, −1.15692618065695592627990638267, 1.72104025056387935453142538470, 2.70109591810614696489657140602, 3.50176973747354530177423554962, 5.04859393260207132524417408755, 6.00227938316500534528149235217, 6.68471276727359786351287976593, 7.70952505944263195648309746557, 8.881042157139473401981444831058, 9.368696572328290259056706036640, 9.891338250008906287884606335217

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