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Properties

Label 2-867-17.13-c1-0-32
Degree $2$
Conductor $867$
Sign $0.698 + 0.715i$
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  − 0.765i·2-s + (0.707 + 0.707i)3-s + 1.41·4-s + (−0.0582 − 0.0582i)5-s + (0.541 − 0.541i)6-s + (1.95 − 1.95i)7-s − 2.61i·8-s + 1.00i·9-s + (−0.0445 + 0.0445i)10-s + (−0.472 + 0.472i)11-s + (1 + i)12-s + 3.28·13-s + (−1.49 − 1.49i)14-s − 0.0823i·15-s + 0.828·16-s + ⋯
L(s)  = 1  − 0.541i·2-s + (0.408 + 0.408i)3-s + 0.707·4-s + (−0.0260 − 0.0260i)5-s + (0.220 − 0.220i)6-s + (0.739 − 0.739i)7-s − 0.923i·8-s + 0.333i·9-s + (−0.0141 + 0.0141i)10-s + (−0.142 + 0.142i)11-s + (0.288 + 0.288i)12-s + 0.910·13-s + (−0.399 − 0.399i)14-s − 0.0212i·15-s + 0.207·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 + 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.698 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $0.698 + 0.715i$
Analytic conductor: \(6.92302\)
Root analytic conductor: \(2.63116\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :1/2),\ 0.698 + 0.715i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.19099 - 0.923620i\)
\(L(\frac12)\) \(\approx\) \(2.19099 - 0.923620i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.707 - 0.707i)T \)
17 \( 1 \)
good2 \( 1 + 0.765iT - 2T^{2} \)
5 \( 1 + (0.0582 + 0.0582i)T + 5iT^{2} \)
7 \( 1 + (-1.95 + 1.95i)T - 7iT^{2} \)
11 \( 1 + (0.472 - 0.472i)T - 11iT^{2} \)
13 \( 1 - 3.28T + 13T^{2} \)
19 \( 1 + 3.64iT - 19T^{2} \)
23 \( 1 + (6.58 - 6.58i)T - 23iT^{2} \)
29 \( 1 + (-4.41 - 4.41i)T + 29iT^{2} \)
31 \( 1 + (3.56 + 3.56i)T + 31iT^{2} \)
37 \( 1 + (1.70 + 1.70i)T + 37iT^{2} \)
41 \( 1 + (0.339 - 0.339i)T - 41iT^{2} \)
43 \( 1 - 8.27iT - 43T^{2} \)
47 \( 1 - 8.88T + 47T^{2} \)
53 \( 1 + 11.3iT - 53T^{2} \)
59 \( 1 - 9.59iT - 59T^{2} \)
61 \( 1 + (1.82 - 1.82i)T - 61iT^{2} \)
67 \( 1 + 0.944T + 67T^{2} \)
71 \( 1 + (-2.32 - 2.32i)T + 71iT^{2} \)
73 \( 1 + (-11.0 - 11.0i)T + 73iT^{2} \)
79 \( 1 + (-5.91 + 5.91i)T - 79iT^{2} \)
83 \( 1 + 0.899iT - 83T^{2} \)
89 \( 1 + 5.64T + 89T^{2} \)
97 \( 1 + (7.75 + 7.75i)T + 97iT^{2} \)
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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

−10.22922833964346876628429635940, −9.419875414289758946008583890624, −8.308463903774699919101916417014, −7.61435923960416688737484942320, −6.72052829894670637882734082418, −5.62106544794280629486675658441, −4.34093364322214796285205971976, −3.62362928261277679685482522496, −2.42841130222436896292885273416, −1.27654562213331550814439556268, 1.64833278688601214016250950196, 2.54301662188662547141164053235, 3.83870682825965808457179666801, 5.29223316103068875082108001586, 6.04018220192507053996562304327, 6.79435698169196807133400965271, 7.940404603963360172625431115107, 8.241525505470663649865822329143, 9.095965566511845116698495125108, 10.44129576242150646714977604237

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