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Properties

Label 2-1045-209.208-c1-0-75
Degree $2$
Conductor $1045$
Sign $-0.999 + 0.0172i$
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  + 1.23·2-s − 3.34i·3-s − 0.469·4-s + 5-s − 4.13i·6-s − 3.14i·7-s − 3.05·8-s − 8.18·9-s + 1.23·10-s + (3.08 − 1.21i)11-s + 1.57i·12-s + 4.68·13-s − 3.89i·14-s − 3.34i·15-s − 2.84·16-s − 0.133i·17-s + ⋯
L(s)  = 1  + 0.874·2-s − 1.93i·3-s − 0.234·4-s + 0.447·5-s − 1.68i·6-s − 1.19i·7-s − 1.08·8-s − 2.72·9-s + 0.391·10-s + (0.929 − 0.367i)11-s + 0.453i·12-s + 1.29·13-s − 1.04i·14-s − 0.863i·15-s − 0.710·16-s − 0.0324i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0172i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.999 + 0.0172i$
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: \((2,\ 1045,\ (\ :1/2),\ -0.999 + 0.0172i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.977623871\)
\(L(\frac12)\) \(\approx\) \(1.977623871\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.08 + 1.21i)T \)
19 \( 1 + (4.08 + 1.53i)T \)
good2 \( 1 - 1.23T + 2T^{2} \)
3 \( 1 + 3.34iT - 3T^{2} \)
7 \( 1 + 3.14iT - 7T^{2} \)
13 \( 1 - 4.68T + 13T^{2} \)
17 \( 1 + 0.133iT - 17T^{2} \)
23 \( 1 - 2.70T + 23T^{2} \)
29 \( 1 + 9.38T + 29T^{2} \)
31 \( 1 - 9.11iT - 31T^{2} \)
37 \( 1 + 6.98iT - 37T^{2} \)
41 \( 1 - 0.185T + 41T^{2} \)
43 \( 1 + 1.06iT - 43T^{2} \)
47 \( 1 - 7.91T + 47T^{2} \)
53 \( 1 - 1.35iT - 53T^{2} \)
59 \( 1 - 3.19iT - 59T^{2} \)
61 \( 1 - 2.28iT - 61T^{2} \)
67 \( 1 + 0.902iT - 67T^{2} \)
71 \( 1 + 13.3iT - 71T^{2} \)
73 \( 1 + 11.2iT - 73T^{2} \)
79 \( 1 - 2.27T + 79T^{2} \)
83 \( 1 - 9.64iT - 83T^{2} \)
89 \( 1 - 9.59iT - 89T^{2} \)
97 \( 1 + 14.1iT - 97T^{2} \)
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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

−9.005625872340884021193456486043, −8.768990427565700982741285359579, −7.55845607049403297498068833901, −6.77634901197390524134298364517, −6.20377950355357739742231919162, −5.46355274105085092397981471452, −4.03647843594719256848199548882, −3.19245670054674015682007841860, −1.73848486371945894985947682641, −0.69034707662340876684013340765, 2.45172500875130992745014963203, 3.63299577249032560190249925186, 4.09714909201369167298138717296, 5.05894910677663665896543391414, 5.83802712426672932981830223689, 6.21079277150235445427199578462, 8.384404302927295056995095885876, 9.011975499623676855208926198578, 9.359738536533910426051055099155, 10.19197201014726967893519358120

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