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Properties

Label 2-891-99.41-c1-0-12
Degree $2$
Conductor $891$
Sign $0.205 + 0.978i$
Analytic cond. $7.11467$
Root an. cond. $2.66733$
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  + (−2.22 − 0.990i)2-s + (2.63 + 2.92i)4-s + (−1.54 − 3.46i)5-s + (0.0764 + 0.359i)7-s + (−1.45 − 4.48i)8-s + 9.24i·10-s + (0.223 + 3.30i)11-s + (0.943 − 0.0991i)13-s + (0.186 − 0.876i)14-s + (−0.378 + 3.59i)16-s + (2.77 + 2.01i)17-s + (4.05 − 1.31i)19-s + (6.07 − 13.6i)20-s + (2.78 − 7.58i)22-s + (4.30 + 2.48i)23-s + ⋯
L(s)  = 1  + (−1.57 − 0.700i)2-s + (1.31 + 1.46i)4-s + (−0.690 − 1.55i)5-s + (0.0289 + 0.135i)7-s + (−0.515 − 1.58i)8-s + 2.92i·10-s + (0.0674 + 0.997i)11-s + (0.261 − 0.0274i)13-s + (0.0497 − 0.234i)14-s + (−0.0945 + 0.899i)16-s + (0.673 + 0.489i)17-s + (0.929 − 0.302i)19-s + (1.35 − 3.05i)20-s + (0.592 − 1.61i)22-s + (0.897 + 0.517i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.205 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.205 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $0.205 + 0.978i$
Analytic conductor: \(7.11467\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ 0.205 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.498150 - 0.404473i\)
\(L(\frac12)\) \(\approx\) \(0.498150 - 0.404473i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + (-0.223 - 3.30i)T \)
good2 \( 1 + (2.22 + 0.990i)T + (1.33 + 1.48i)T^{2} \)
5 \( 1 + (1.54 + 3.46i)T + (-3.34 + 3.71i)T^{2} \)
7 \( 1 + (-0.0764 - 0.359i)T + (-6.39 + 2.84i)T^{2} \)
13 \( 1 + (-0.943 + 0.0991i)T + (12.7 - 2.70i)T^{2} \)
17 \( 1 + (-2.77 - 2.01i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-4.05 + 1.31i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + (-4.30 - 2.48i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.42 + 0.516i)T + (26.4 - 11.7i)T^{2} \)
31 \( 1 + (0.367 + 3.49i)T + (-30.3 + 6.44i)T^{2} \)
37 \( 1 + (2.21 - 6.83i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.65 - 0.565i)T + (37.4 + 16.6i)T^{2} \)
43 \( 1 + (-1.63 + 0.943i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.0153 + 0.0137i)T + (4.91 + 46.7i)T^{2} \)
53 \( 1 + (3.25 + 4.48i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (4.92 - 4.43i)T + (6.16 - 58.6i)T^{2} \)
61 \( 1 + (-9.69 - 1.01i)T + (59.6 + 12.6i)T^{2} \)
67 \( 1 + (2.23 - 3.86i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.06 + 8.35i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (4.18 + 1.35i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-4.44 + 9.99i)T + (-52.8 - 58.7i)T^{2} \)
83 \( 1 + (-0.959 + 9.13i)T + (-81.1 - 17.2i)T^{2} \)
89 \( 1 - 3.04iT - 89T^{2} \)
97 \( 1 + (13.7 + 6.13i)T + (64.9 + 72.0i)T^{2} \)
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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

−9.674692779064506389514564925595, −9.206003850280788826184032450363, −8.417231870919477435015498712222, −7.81495224330888969409870960569, −7.06513104945297336081604527951, −5.44132160991668579718044920688, −4.46370513298519560290842476181, −3.23357467423821227612927833545, −1.71215402225324482371077807215, −0.799082286957592673490307948801, 0.877545975192207856243061107422, 2.73016554365286383156725595667, 3.67586602866117347508832031244, 5.52789742399089835748987599461, 6.46323874658651931191284715899, 7.14075881372553392450059768661, 7.71425998068751741202345375307, 8.502173665418391664655076075552, 9.393177571529614932816895912878, 10.26296204158724030320327626731

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