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Properties

Label 2-1323-63.5-c1-0-28
Degree $2$
Conductor $1323$
Sign $0.135 + 0.990i$
Analytic cond. $10.5642$
Root an. cond. $3.25026$
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  + 0.641i·2-s + 1.58·4-s + (−1.10 − 1.91i)5-s + 2.30i·8-s + (1.22 − 0.709i)10-s + (−2.93 − 1.69i)11-s + (−1.56 − 0.901i)13-s + 1.69·16-s + (−2.98 − 5.16i)17-s + (1.42 + 0.822i)19-s + (−1.75 − 3.04i)20-s + (1.08 − 1.88i)22-s + (2.05 − 1.18i)23-s + (0.0556 − 0.0963i)25-s + (0.578 − 1.00i)26-s + ⋯
L(s)  = 1  + 0.453i·2-s + 0.794·4-s + (−0.494 − 0.856i)5-s + 0.813i·8-s + (0.388 − 0.224i)10-s + (−0.885 − 0.511i)11-s + (−0.432 − 0.249i)13-s + 0.424·16-s + (−0.723 − 1.25i)17-s + (0.326 + 0.188i)19-s + (−0.392 − 0.680i)20-s + (0.232 − 0.401i)22-s + (0.428 − 0.247i)23-s + (0.0111 − 0.0192i)25-s + (0.113 − 0.196i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.135 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.135 + 0.990i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1097, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.135 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.287467215\)
\(L(\frac12)\) \(\approx\) \(1.287467215\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 - 0.641iT - 2T^{2} \)
5 \( 1 + (1.10 + 1.91i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.93 + 1.69i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.56 + 0.901i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.98 + 5.16i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.42 - 0.822i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.05 + 1.18i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.44 - 1.41i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 10.7iT - 31T^{2} \)
37 \( 1 + (0.849 - 1.47i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.455 + 0.788i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.96 + 3.39i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.246T + 47T^{2} \)
53 \( 1 + (-6.82 + 3.93i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 - 1.41iT - 61T^{2} \)
67 \( 1 + 7.98T + 67T^{2} \)
71 \( 1 + 12.1iT - 71T^{2} \)
73 \( 1 + (-0.369 + 0.213i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 4.98T + 79T^{2} \)
83 \( 1 + (-4.28 - 7.42i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.26 + 9.12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.30 - 3.63i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.270025333204962937658456090602, −8.472909562136739714139458706654, −7.71235497765338841540104674018, −7.20066535194150633635223468048, −6.09284655047091435258740078864, −5.27682115700268417529250357935, −4.56724694648442277860209629498, −3.15272776438664410417859269950, −2.20843874768207359091829018669, −0.49150451625913265953908348141, 1.67195414296029963960180952817, 2.71156985794514929222195022035, 3.44925815300090062996728005051, 4.56779433122907776677997300412, 5.74871276212604879349738613357, 6.79955355700012677883589227314, 7.21513455904322043757396916356, 8.022908319727111043661306931813, 9.116958296211836332077590301529, 10.17359019967780631318296781009

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