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Properties

Label 2-1045-5.4-c1-0-12
Degree $2$
Conductor $1045$
Sign $-0.786 + 0.617i$
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  + 0.661i·2-s + 3.01i·3-s + 1.56·4-s + (−1.75 + 1.38i)5-s − 1.99·6-s − 0.592i·7-s + 2.35i·8-s − 6.06·9-s + (−0.913 − 1.16i)10-s + 11-s + 4.70i·12-s + 3.68i·13-s + 0.391·14-s + (−4.16 − 5.29i)15-s + 1.56·16-s − 0.251i·17-s + ⋯
L(s)  = 1  + 0.467i·2-s + 1.73i·3-s + 0.781·4-s + (−0.786 + 0.617i)5-s − 0.812·6-s − 0.223i·7-s + 0.832i·8-s − 2.02·9-s + (−0.288 − 0.367i)10-s + 0.301·11-s + 1.35i·12-s + 1.02i·13-s + 0.104·14-s + (−1.07 − 1.36i)15-s + 0.391·16-s − 0.0608i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.786 + 0.617i$
Analytic conductor: \(8.34436\)
Root analytic conductor: \(2.88866\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :1/2),\ -0.786 + 0.617i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.378175530\)
\(L(\frac12)\) \(\approx\) \(1.378175530\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.75 - 1.38i)T \)
11 \( 1 - T \)
19 \( 1 + T \)
good2 \( 1 - 0.661iT - 2T^{2} \)
3 \( 1 - 3.01iT - 3T^{2} \)
7 \( 1 + 0.592iT - 7T^{2} \)
13 \( 1 - 3.68iT - 13T^{2} \)
17 \( 1 + 0.251iT - 17T^{2} \)
23 \( 1 - 2.58iT - 23T^{2} \)
29 \( 1 + 2.22T + 29T^{2} \)
31 \( 1 + 7.50T + 31T^{2} \)
37 \( 1 + 9.64iT - 37T^{2} \)
41 \( 1 - 5.78T + 41T^{2} \)
43 \( 1 + 0.283iT - 43T^{2} \)
47 \( 1 - 3.78iT - 47T^{2} \)
53 \( 1 - 3.89iT - 53T^{2} \)
59 \( 1 - 2.98T + 59T^{2} \)
61 \( 1 + 8.94T + 61T^{2} \)
67 \( 1 + 3.70iT - 67T^{2} \)
71 \( 1 - 16.1T + 71T^{2} \)
73 \( 1 - 7.91iT - 73T^{2} \)
79 \( 1 - 15.4T + 79T^{2} \)
83 \( 1 - 3.74iT - 83T^{2} \)
89 \( 1 - 3.78T + 89T^{2} \)
97 \( 1 + 9.18iT - 97T^{2} \)
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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

−10.74343653414965857985942200374, −9.521839803881899259729019799766, −8.956691360464205455910519084329, −7.83439138629536644435671803420, −7.10859804908994036627036903011, −6.14716234904639030137218180446, −5.22902322437346041932601234376, −4.10272587934800168409431794527, −3.60438763889070930401840855976, −2.37334803281583519758729060847, 0.60185009238274549874775508301, 1.63405097360232601883379123924, 2.68280663452872490602815712043, 3.70013735944685115483191230153, 5.30183414691155762092865443050, 6.23291200988777786408701984857, 7.00162708503752950919435548620, 7.74330727316950185007715596916, 8.265173996420692017482703679835, 9.249671000614725916995265817165

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