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Properties

Label 2-867-17.13-c1-0-1
Degree $2$
Conductor $867$
Sign $-0.563 - 0.825i$
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  − 0.907i·2-s + (0.707 + 0.707i)3-s + 1.17·4-s + (−2.25 − 2.25i)5-s + (0.641 − 0.641i)6-s + (−2.51 + 2.51i)7-s − 2.88i·8-s + 1.00i·9-s + (−2.04 + 2.04i)10-s + (−2.31 + 2.31i)11-s + (0.832 + 0.832i)12-s − 5.58·13-s + (2.28 + 2.28i)14-s − 3.19i·15-s − 0.259·16-s + ⋯
L(s)  = 1  − 0.641i·2-s + (0.408 + 0.408i)3-s + 0.588·4-s + (−1.00 − 1.00i)5-s + (0.261 − 0.261i)6-s + (−0.952 + 0.952i)7-s − 1.01i·8-s + 0.333i·9-s + (−0.647 + 0.647i)10-s + (−0.698 + 0.698i)11-s + (0.240 + 0.240i)12-s − 1.54·13-s + (0.610 + 0.610i)14-s − 0.824i·15-s − 0.0649·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 - 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.563 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $-0.563 - 0.825i$
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: \((2,\ 867,\ (\ :1/2),\ -0.563 - 0.825i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.121076 + 0.229308i\)
\(L(\frac12)\) \(\approx\) \(0.121076 + 0.229308i\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.907iT - 2T^{2} \)
5 \( 1 + (2.25 + 2.25i)T + 5iT^{2} \)
7 \( 1 + (2.51 - 2.51i)T - 7iT^{2} \)
11 \( 1 + (2.31 - 2.31i)T - 11iT^{2} \)
13 \( 1 + 5.58T + 13T^{2} \)
19 \( 1 - 4.23iT - 19T^{2} \)
23 \( 1 + (3.25 - 3.25i)T - 23iT^{2} \)
29 \( 1 + (1.47 + 1.47i)T + 29iT^{2} \)
31 \( 1 + (-0.317 - 0.317i)T + 31iT^{2} \)
37 \( 1 + (0.525 + 0.525i)T + 37iT^{2} \)
41 \( 1 + (-3.17 + 3.17i)T - 41iT^{2} \)
43 \( 1 - 6.10iT - 43T^{2} \)
47 \( 1 - 2.26T + 47T^{2} \)
53 \( 1 + 7.55iT - 53T^{2} \)
59 \( 1 + 2.83iT - 59T^{2} \)
61 \( 1 + (-2.76 + 2.76i)T - 61iT^{2} \)
67 \( 1 + 14.5T + 67T^{2} \)
71 \( 1 + (2.57 + 2.57i)T + 71iT^{2} \)
73 \( 1 + (8.24 + 8.24i)T + 73iT^{2} \)
79 \( 1 + (3.98 - 3.98i)T - 79iT^{2} \)
83 \( 1 - 3.92iT - 83T^{2} \)
89 \( 1 + 14.6T + 89T^{2} \)
97 \( 1 + (-0.585 - 0.585i)T + 97iT^{2} \)
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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

−10.17504864730190404601547380059, −9.789060410546449539482015992249, −8.978010256908655224925430601340, −7.86132144574287814465775340654, −7.36518003313214036026710896958, −5.98420759656495110970260415231, −4.94308117406455134452551972256, −3.95539754338994997340776646281, −2.95685940128622915825725318007, −2.01966151259195377711850202057, 0.10476243078996922522992427784, 2.59618168438232172897828067185, 3.11145379925149871808222970690, 4.37242899480842680577421819005, 5.83031174577934551269378344021, 6.82871407879180596567921792096, 7.27999170765376608002059185945, 7.69418841671638970443540867071, 8.739304727189023705296353045429, 10.07686786701912778325256660154

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