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Properties

Label 2-867-17.9-c1-0-25
Degree $2$
Conductor $867$
Sign $0.993 - 0.113i$
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  + (1.10 − 1.10i)2-s + (0.923 + 0.382i)3-s − 0.438i·4-s + (−0.214 + 0.518i)5-s + (1.44 − 0.597i)6-s + (1.72 + 1.72i)8-s + (0.707 + 0.707i)9-s + (0.335 + 0.810i)10-s + (−2.36 + 0.980i)11-s + (0.167 − 0.405i)12-s + 4.56i·13-s + (−0.397 + 0.397i)15-s + 4.68·16-s + 1.56·18-s + (5.43 − 5.43i)19-s + (0.227 + 0.0942i)20-s + ⋯
L(s)  = 1  + (0.780 − 0.780i)2-s + (0.533 + 0.220i)3-s − 0.219i·4-s + (−0.0961 + 0.232i)5-s + (0.588 − 0.243i)6-s + (0.609 + 0.609i)8-s + (0.235 + 0.235i)9-s + (0.106 + 0.256i)10-s + (−0.713 + 0.295i)11-s + (0.0484 − 0.116i)12-s + 1.26i·13-s + (−0.102 + 0.102i)15-s + 1.17·16-s + 0.368·18-s + (1.24 − 1.24i)19-s + (0.0508 + 0.0210i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.113i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $0.993 - 0.113i$
Analytic conductor: \(6.92302\)
Root analytic conductor: \(2.63116\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :1/2),\ 0.993 - 0.113i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.83451 + 0.160726i\)
\(L(\frac12)\) \(\approx\) \(2.83451 + 0.160726i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.10 + 1.10i)T - 2iT^{2} \)
5 \( 1 + (0.214 - 0.518i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (2.36 - 0.980i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 - 4.56iT - 13T^{2} \)
19 \( 1 + (-5.43 + 5.43i)T - 19iT^{2} \)
23 \( 1 + (-6.06 + 2.51i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (3.15 - 7.61i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (4.73 + 1.96i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (2.88 + 1.19i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-0.214 - 0.518i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (5.43 + 5.43i)T + 43iT^{2} \)
47 \( 1 + 2.87iT - 47T^{2} \)
53 \( 1 + (3.00 - 3.00i)T - 53iT^{2} \)
59 \( 1 + (-0.794 - 0.794i)T + 59iT^{2} \)
61 \( 1 + (0.335 + 0.810i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + (9.46 + 3.92i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-1.62 + 3.92i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-14.1 + 5.88i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (-6.45 + 6.45i)T - 83iT^{2} \)
89 \( 1 + 7.12iT - 89T^{2} \)
97 \( 1 + (-4.25 + 10.2i)T + (-68.5 - 68.5i)T^{2} \)
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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

−10.49169006184977390515835974427, −9.258341751610189974835039622181, −8.769651687804051757072459892795, −7.39060889912627021339743516348, −7.03709057175626455753077534453, −5.23517023459594071597168382740, −4.75212234543252015506528388975, −3.56061530053079638434001542838, −2.87980596366055077394745112250, −1.78410571338470821777466263478, 1.14094855068135874819642341839, 2.92578377917436437176980993039, 3.82200038872713301942425801659, 5.13039060726882335939941701084, 5.58264360942751630288496319529, 6.63802736261225620004253133430, 7.76250696337635457446448078125, 7.930757629229980796454194295868, 9.255299164653169606002970254141, 10.11412382926631848453995380888

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