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Properties

Label 2-921-921.263-c1-0-53
Degree $2$
Conductor $921$
Sign $0.794 + 0.607i$
Analytic cond. $7.35422$
Root an. cond. $2.71186$
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  + (−1.38 + 1.04i)3-s + (0.0615 + 1.99i)4-s + (−3.06 − 0.252i)7-s + (0.820 − 2.88i)9-s + (−2.17 − 2.69i)12-s + (1.88 − 6.37i)13-s + (−3.99 + 0.246i)16-s + (−1.30 − 1.97i)19-s + (4.50 − 2.85i)21-s + (3.62 + 3.44i)25-s + (1.87 + 4.84i)27-s + (0.316 − 6.15i)28-s + (−2.39 − 10.4i)31-s + (5.81 + 1.46i)36-s + (8.63 + 8.54i)37-s + ⋯
L(s)  = 1  + (−0.798 + 0.602i)3-s + (0.0307 + 0.999i)4-s + (−1.16 − 0.0955i)7-s + (0.273 − 0.961i)9-s + (−0.626 − 0.779i)12-s + (0.523 − 1.76i)13-s + (−0.998 + 0.0615i)16-s + (−0.300 − 0.452i)19-s + (0.983 − 0.623i)21-s + (0.725 + 0.688i)25-s + (0.361 + 0.932i)27-s + (0.0597 − 1.16i)28-s + (−0.429 − 1.87i)31-s + (0.969 + 0.243i)36-s + (1.41 + 1.40i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(921\)    =    \(3 \cdot 307\)
Sign: $0.794 + 0.607i$
Analytic conductor: \(7.35422\)
Root analytic conductor: \(2.71186\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{921} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 921,\ (\ :1/2),\ 0.794 + 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.668686 - 0.226458i\)
\(L(\frac12)\) \(\approx\) \(0.668686 - 0.226458i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.38 - 1.04i)T \)
307 \( 1 + (-9.85 - 14.4i)T \)
good2 \( 1 + (-0.0615 - 1.99i)T^{2} \)
5 \( 1 + (-3.62 - 3.44i)T^{2} \)
7 \( 1 + (3.06 + 0.252i)T + (6.90 + 1.14i)T^{2} \)
11 \( 1 + (10.9 - 1.12i)T^{2} \)
13 \( 1 + (-1.88 + 6.37i)T + (-10.9 - 7.06i)T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.30 + 1.97i)T + (-7.40 + 17.4i)T^{2} \)
23 \( 1 + (14.2 + 18.0i)T^{2} \)
29 \( 1 + (-9.07 + 27.5i)T^{2} \)
31 \( 1 + (2.39 + 10.4i)T + (-27.8 + 13.5i)T^{2} \)
37 \( 1 + (-8.63 - 8.54i)T + (0.379 + 36.9i)T^{2} \)
41 \( 1 + (-35.2 - 20.8i)T^{2} \)
43 \( 1 + (-0.146 + 3.57i)T + (-42.8 - 3.52i)T^{2} \)
47 \( 1 + (13.7 + 44.9i)T^{2} \)
53 \( 1 + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (30.5 + 50.4i)T^{2} \)
61 \( 1 + (-0.776 + 1.88i)T + (-43.3 - 42.9i)T^{2} \)
67 \( 1 + (3.08 + 9.05i)T + (-53.0 + 40.9i)T^{2} \)
71 \( 1 + (40.4 + 58.3i)T^{2} \)
73 \( 1 + (-12.8 + 3.08i)T + (65.0 - 33.2i)T^{2} \)
79 \( 1 + (12.1 + 3.87i)T + (64.4 + 45.6i)T^{2} \)
83 \( 1 + (-0.852 + 82.9i)T^{2} \)
89 \( 1 + (82.3 + 33.8i)T^{2} \)
97 \( 1 + (-7.57 + 11.4i)T + (-37.8 - 89.3i)T^{2} \)
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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

−9.981201936714408487926679108301, −9.318951094495461959812414310841, −8.332644080318318555145404259789, −7.40631652439408859638285578515, −6.45508281182822604935966160853, −5.74013894037065815214272385585, −4.57903345066076997453249356528, −3.54534447798825841307550593730, −2.92400627505529718190800780918, −0.41747861508490617642254098006, 1.18446978962972229642569781640, 2.36647120324374489306998964776, 4.05457594633418877539876840993, 5.07183904886467079529642137470, 6.15803710385908404708304729312, 6.48691674245595713174349687347, 7.22933065428886469021827533357, 8.692755424722376199185003879924, 9.420544222381094537093208408312, 10.27384604337062796350784183084

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