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Properties

Label 2-867-17.15-c1-0-35
Degree $2$
Conductor $867$
Sign $0.691 - 0.722i$
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 + (0.382 + 0.923i)3-s + 0.729i·4-s + (3.75 − 1.55i)5-s + (−0.632 + 1.52i)6-s + (−0.852 − 0.352i)7-s + (1.48 − 1.48i)8-s + (−0.707 + 0.707i)9-s + (6.20 + 2.57i)10-s + (−0.868 + 2.09i)11-s + (−0.674 + 0.279i)12-s − 3.57i·13-s + (−0.583 − 1.40i)14-s + (2.87 + 2.87i)15-s + 4.92·16-s + ⋯
L(s)  = 1  + (0.826 + 0.826i)2-s + (0.220 + 0.533i)3-s + 0.364i·4-s + (1.68 − 0.695i)5-s + (−0.258 + 0.623i)6-s + (−0.322 − 0.133i)7-s + (0.524 − 0.524i)8-s + (−0.235 + 0.235i)9-s + (1.96 + 0.813i)10-s + (−0.261 + 0.632i)11-s + (−0.194 + 0.0806i)12-s − 0.991i·13-s + (−0.155 − 0.376i)14-s + (0.742 + 0.742i)15-s + 1.23·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $0.691 - 0.722i$
Analytic conductor: \(6.92302\)
Root analytic conductor: \(2.63116\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (712, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :1/2),\ 0.691 - 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.05708 + 1.30532i\)
\(L(\frac12)\) \(\approx\) \(3.05708 + 1.30532i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.16 - 1.16i)T + 2iT^{2} \)
5 \( 1 + (-3.75 + 1.55i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (0.852 + 0.352i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (0.868 - 2.09i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + 3.57iT - 13T^{2} \)
19 \( 1 + (1.22 + 1.22i)T + 19iT^{2} \)
23 \( 1 + (1.95 - 4.71i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (2.03 - 0.843i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-1.58 - 3.81i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (-2.27 - 5.50i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (4.66 + 1.93i)T + (28.9 + 28.9i)T^{2} \)
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 \( 1 + (3.59 + 1.48i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + 12.0T + 67T^{2} \)
71 \( 1 + (-0.583 - 1.40i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (1.07 - 0.445i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (4.54 - 10.9i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (4.51 + 4.51i)T + 83iT^{2} \)
89 \( 1 - 10.3iT - 89T^{2} \)
97 \( 1 + (13.3 - 5.53i)T + (68.5 - 68.5i)T^{2} \)
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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

−10.02962253747010555122673263470, −9.626491135145719411252013369465, −8.635445883265140055282123740854, −7.53727897552183604380813827755, −6.49786410095184988692942325253, −5.68463598910218600775697614614, −5.18341439805411266411959156354, −4.34475535363641663224924336429, −2.95987999573746932645652821914, −1.52549536638230159480658428587, 1.76268593477624689996043073039, 2.43566261203943532605639123562, 3.28436405352741701293887779608, 4.56167338161440219575321727316, 5.83543731470855920011069913442, 6.24873687581527638637656817679, 7.31898898963727662799763489720, 8.494657986629506393077213207951, 9.415287995434243245690786979967, 10.23477220752817822609348384726

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