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Properties

Label 2-867-17.9-c1-0-30
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  + (1.46 − 1.46i)2-s + (−0.923 − 0.382i)3-s − 2.27i·4-s + (0.133 − 0.321i)5-s + (−1.90 + 0.790i)6-s + (1.65 + 4.00i)7-s + (−0.394 − 0.394i)8-s + (0.707 + 0.707i)9-s + (−0.275 − 0.663i)10-s + (0.674 − 0.279i)11-s + (−0.868 + 2.09i)12-s − 5.40i·13-s + (8.27 + 3.42i)14-s + (−0.245 + 0.245i)15-s + 3.38·16-s + ⋯
L(s)  = 1  + (1.03 − 1.03i)2-s + (−0.533 − 0.220i)3-s − 1.13i·4-s + (0.0595 − 0.143i)5-s + (−0.779 + 0.322i)6-s + (0.627 + 1.51i)7-s + (−0.139 − 0.139i)8-s + (0.235 + 0.235i)9-s + (−0.0869 − 0.209i)10-s + (0.203 − 0.0842i)11-s + (−0.250 + 0.605i)12-s − 1.49i·13-s + (2.21 + 0.916i)14-s + (−0.0634 + 0.0634i)15-s + 0.846·16-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: $\chi_{867} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :1/2),\ 0.299 + 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.10475 - 1.54519i\)
\(L(\frac12)\) \(\approx\) \(2.10475 - 1.54519i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.46 + 1.46i)T - 2iT^{2} \)
5 \( 1 + (-0.133 + 0.321i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (-1.65 - 4.00i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-0.674 + 0.279i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + 5.40iT - 13T^{2} \)
19 \( 1 + (-2.31 + 2.31i)T - 19iT^{2} \)
23 \( 1 + (-3.28 + 1.36i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (1.67 - 4.04i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (-3.05 - 1.26i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (2.65 + 1.09i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (4.53 + 10.9i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-2.64 - 2.64i)T + 43iT^{2} \)
47 \( 1 - 0.476iT - 47T^{2} \)
53 \( 1 + (7.16 - 7.16i)T - 53iT^{2} \)
59 \( 1 + (3.74 + 3.74i)T + 59iT^{2} \)
61 \( 1 + (-1.35 - 3.27i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 - 1.55T + 67T^{2} \)
71 \( 1 + (8.27 + 3.42i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-1.56 + 3.78i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (0.750 - 0.310i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (5.10 - 5.10i)T - 83iT^{2} \)
89 \( 1 - 9.77iT - 89T^{2} \)
97 \( 1 + (0.678 - 1.63i)T + (-68.5 - 68.5i)T^{2} \)
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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

−10.49999548658432451133022299453, −9.232820355972761740259656824273, −8.428497214259981822655082878162, −7.38355361522734970762441869727, −5.99939740600368179843900279657, −5.24127369952245329366974647498, −4.90848828685536454904213605095, −3.33290534412193318491541766837, −2.53715858179382792626193508593, −1.30013283965300081595902641062, 1.34623462694046095696230842885, 3.53180759803479182997351514726, 4.44435349316174834163971676539, 4.80398982810193808571258906862, 6.06387202497726503632675019745, 6.80824681936021698720452695867, 7.33086119908269891102455199916, 8.272794510334179411100005589638, 9.646870986092822833673163167200, 10.35691026858653088269374457190

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