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Properties

Label 2-1323-9.4-c1-0-1
Degree $2$
Conductor $1323$
Sign $0.0910 - 0.995i$
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  + (−0.551 − 0.955i)2-s + (0.391 − 0.678i)4-s + (0.0527 − 0.0913i)5-s − 3.07·8-s − 0.116·10-s + (1.66 + 2.89i)11-s + (−1.23 + 2.14i)13-s + (0.909 + 1.57i)16-s − 1.61·17-s − 7.68·19-s + (−0.0413 − 0.0715i)20-s + (1.84 − 3.18i)22-s + (−0.948 + 1.64i)23-s + (2.49 + 4.32i)25-s + 2.73·26-s + ⋯
L(s)  = 1  + (−0.389 − 0.675i)2-s + (0.195 − 0.339i)4-s + (0.0235 − 0.0408i)5-s − 1.08·8-s − 0.0367·10-s + (0.503 + 0.871i)11-s + (−0.343 + 0.595i)13-s + (0.227 + 0.393i)16-s − 0.391·17-s − 1.76·19-s + (−0.00924 − 0.0160i)20-s + (0.392 − 0.679i)22-s + (−0.197 + 0.342i)23-s + (0.498 + 0.864i)25-s + 0.536·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0910 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0910 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.0910 - 0.995i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (442, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.0910 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3894170308\)
\(L(\frac12)\) \(\approx\) \(0.3894170308\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.551 + 0.955i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.0527 + 0.0913i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.66 - 2.89i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.23 - 2.14i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.61T + 17T^{2} \)
19 \( 1 + 7.68T + 19T^{2} \)
23 \( 1 + (0.948 - 1.64i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.64 + 8.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.63 - 8.02i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.98T + 37T^{2} \)
41 \( 1 + (3.74 - 6.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.77 + 6.53i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.59 + 2.76i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 9.97T + 53T^{2} \)
59 \( 1 + (-2.22 + 3.86i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.83 - 4.91i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.98 - 8.63i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.29T + 71T^{2} \)
73 \( 1 + 4.72T + 73T^{2} \)
79 \( 1 + (3.84 + 6.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.584 - 1.01i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6.02T + 89T^{2} \)
97 \( 1 + (1.90 + 3.29i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.972662140770834928249508451818, −9.070345686449300502728570662862, −8.615542724285376401301719565811, −7.20200187044741929197084781071, −6.66602050162052632592654307549, −5.69272692623669959175768260358, −4.63176996231828623217932039557, −3.65350525598937401662741103833, −2.27525485240825416099002785001, −1.63525512824866788971537095449, 0.16790538681243046143846023998, 2.17324657440973881614896763845, 3.27262540614356541505369720233, 4.23035349805692837712497343117, 5.54593620207844509443538105098, 6.31812966160982895148460205869, 6.96086459368736949290036381548, 7.86537823442575090052593091031, 8.661996562119417795485194456492, 9.002423308565714025642749165007

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