Properties

Label 2-1045-209.208-c1-0-67
Degree $2$
Conductor $1045$
Sign $-0.998 + 0.0570i$
Analytic cond. $8.34436$
Root an. cond. $2.88866$
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  + 0.233·2-s − 2.00i·3-s − 1.94·4-s + 5-s − 0.467i·6-s + 1.43i·7-s − 0.921·8-s − 1.00·9-s + 0.233·10-s + (2.74 − 1.86i)11-s + 3.89i·12-s − 5.35·13-s + 0.334i·14-s − 2.00i·15-s + 3.67·16-s − 4.91i·17-s + ⋯
L(s)  = 1  + 0.165·2-s − 1.15i·3-s − 0.972·4-s + 0.447·5-s − 0.190i·6-s + 0.541i·7-s − 0.325·8-s − 0.335·9-s + 0.0738·10-s + (0.826 − 0.562i)11-s + 1.12i·12-s − 1.48·13-s + 0.0894i·14-s − 0.516i·15-s + 0.918·16-s − 1.19i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0570i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0570i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.998 + 0.0570i$
Analytic conductor: \(8.34436\)
Root analytic conductor: \(2.88866\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (626, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :1/2),\ -0.998 + 0.0570i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7463105915\)
\(L(\frac12)\) \(\approx\) \(0.7463105915\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
11 \( 1 + (-2.74 + 1.86i)T \)
19 \( 1 + (3.73 + 2.24i)T \)
good2 \( 1 - 0.233T + 2T^{2} \)
3 \( 1 + 2.00iT - 3T^{2} \)
7 \( 1 - 1.43iT - 7T^{2} \)
13 \( 1 + 5.35T + 13T^{2} \)
17 \( 1 + 4.91iT - 17T^{2} \)
23 \( 1 + 4.98T + 23T^{2} \)
29 \( 1 + 4.09T + 29T^{2} \)
31 \( 1 + 3.87iT - 31T^{2} \)
37 \( 1 - 5.67iT - 37T^{2} \)
41 \( 1 + 5.79T + 41T^{2} \)
43 \( 1 - 9.77iT - 43T^{2} \)
47 \( 1 + 9.64T + 47T^{2} \)
53 \( 1 + 6.67iT - 53T^{2} \)
59 \( 1 + 3.93iT - 59T^{2} \)
61 \( 1 + 9.62iT - 61T^{2} \)
67 \( 1 - 12.3iT - 67T^{2} \)
71 \( 1 - 1.57iT - 71T^{2} \)
73 \( 1 - 9.80iT - 73T^{2} \)
79 \( 1 - 1.60T + 79T^{2} \)
83 \( 1 + 8.47iT - 83T^{2} \)
89 \( 1 + 16.0iT - 89T^{2} \)
97 \( 1 + 13.9iT - 97T^{2} \)
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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

−9.571569885805696879194456598229, −8.627135099651427601225879180879, −7.893315909937034167662973555844, −6.90881483066821229090619264165, −6.17449386150867049545444555057, −5.21889270567187233702200578304, −4.35419454403472805363303239880, −2.91633128573560078903178881644, −1.83687521508110468339959845319, −0.31722691555061175144425540385, 1.86528330464251608188620520573, 3.66244968216804581515779754222, 4.15242282430359940329907213220, 4.90747870713527712756936063830, 5.79234811403528345646334039909, 6.92933992661888724906079958089, 8.002172492613445404007331808884, 9.024284954846415228100835509746, 9.542761279138665643624803703464, 10.31363274024432795474576443090

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