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Properties

Label 2-867-17.2-c1-0-3
Degree $2$
Conductor $867$
Sign $0.299 - 0.954i$
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  + (−1.16 − 1.16i)2-s + (−0.923 + 0.382i)3-s + 0.729i·4-s + (1.55 + 3.75i)5-s + (1.52 + 0.632i)6-s + (−0.352 + 0.852i)7-s + (−1.48 + 1.48i)8-s + (0.707 − 0.707i)9-s + (2.57 − 6.20i)10-s + (2.09 + 0.868i)11-s + (−0.279 − 0.674i)12-s − 3.57i·13-s + (1.40 − 0.583i)14-s + (−2.87 − 2.87i)15-s + 4.92·16-s + ⋯
L(s)  = 1  + (−0.826 − 0.826i)2-s + (−0.533 + 0.220i)3-s + 0.364i·4-s + (0.695 + 1.68i)5-s + (0.623 + 0.258i)6-s + (−0.133 + 0.322i)7-s + (−0.524 + 0.524i)8-s + (0.235 − 0.235i)9-s + (0.813 − 1.96i)10-s + (0.632 + 0.261i)11-s + (−0.0806 − 0.194i)12-s − 0.991i·13-s + (0.376 − 0.155i)14-s + (−0.742 − 0.742i)15-s + 1.23·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $0.299 - 0.954i$
Analytic conductor: \(6.92302\)
Root analytic conductor: \(2.63116\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (733, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :1/2),\ 0.299 - 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.601739 + 0.441763i\)
\(L(\frac12)\) \(\approx\) \(0.601739 + 0.441763i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.923 - 0.382i)T \)
17 \( 1 \)
good2 \( 1 + (1.16 + 1.16i)T + 2iT^{2} \)
5 \( 1 + (-1.55 - 3.75i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (0.352 - 0.852i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (-2.09 - 0.868i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + 3.57iT - 13T^{2} \)
19 \( 1 + (-1.22 - 1.22i)T + 19iT^{2} \)
23 \( 1 + (-4.71 - 1.95i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (0.843 + 2.03i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (3.81 - 1.58i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (5.50 - 2.27i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (1.93 - 4.66i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (3.69 - 3.69i)T - 43iT^{2} \)
47 \( 1 - 6.96iT - 47T^{2} \)
53 \( 1 + (1.90 + 1.90i)T + 53iT^{2} \)
59 \( 1 + (-5.23 + 5.23i)T - 59iT^{2} \)
61 \( 1 + (1.48 - 3.59i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + 12.0T + 67T^{2} \)
71 \( 1 + (1.40 - 0.583i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (0.445 + 1.07i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-10.9 - 4.54i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (-4.51 - 4.51i)T + 83iT^{2} \)
89 \( 1 - 10.3iT - 89T^{2} \)
97 \( 1 + (5.53 + 13.3i)T + (-68.5 + 68.5i)T^{2} \)
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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

−10.37579897002668911700965087357, −9.732255274723915370468922693296, −9.138715809703418846647094590646, −7.83892803085689122976985346895, −6.82281279054069686988028505035, −6.04560067269845510088759275685, −5.27723213449767135703947602623, −3.44021566712961940956403634411, −2.73285111721481243458816744151, −1.49439654099981412693485065727, 0.53985718189026232921934315207, 1.68240653170168597063349318833, 3.82968286410505481905203171257, 4.95373916807711372859183809709, 5.75650075311048715870005834281, 6.69401028625216730861595230347, 7.33651296024477493006822336100, 8.611530537368589525517967716374, 8.934646794570823070519311353296, 9.579050093478354427498750884773

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