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Properties

Label 1-1045-1045.569-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.970 - 0.242i$
Analytic cond. $4.85295$
Root an. cond. $4.85295$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 + 0.951i)2-s + (−0.809 + 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)6-s + (−0.809 − 0.587i)7-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + 12-s + (−0.309 + 0.951i)13-s + (0.809 − 0.587i)14-s + (0.309 + 0.951i)16-s + (0.309 + 0.951i)17-s + (0.809 + 0.587i)18-s + 21-s − 23-s + (−0.309 + 0.951i)24-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + (−0.809 + 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)6-s + (−0.809 − 0.587i)7-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + 12-s + (−0.309 + 0.951i)13-s + (0.809 − 0.587i)14-s + (0.309 + 0.951i)16-s + (0.309 + 0.951i)17-s + (0.809 + 0.587i)18-s + 21-s − 23-s + (−0.309 + 0.951i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.970 - 0.242i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.970 - 0.242i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3429665136 - 0.04215966795i\)
\(L(\frac12)\) \(\approx\) \(0.3429665136 - 0.04215966795i\)
\(L(1)\) \(\approx\) \(0.4548600805 + 0.2420944032i\)
\(L(1)\) \(\approx\) \(0.4548600805 + 0.2420944032i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.309 + 0.951i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.809 + 0.587i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 - T \)
97 \( 1 + (0.309 - 0.951i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.81968324476723487957311540586, −20.60082141023963195426225471396, −20.01116218849914423279227453810, −18.987843003971627905963679338312, −18.635814574746116731558818700305, −17.86141851775092489574477261132, −17.12376749057658505741369682850, −16.32615380194318577350363266237, −15.511996133886287766074916217625, −14.14909128986421006927475523696, −13.30705915789712285109866401445, −12.61539973378782782820655544278, −12.082045691385163199127014084988, −11.34894368461731142042730790628, −10.38599893540885432645930863123, −9.78705421868169889618770593989, −8.82418021468702921860105418816, −7.78778916841854888412323373585, −7.068935968254565135860171008192, −5.773692298232011161499322739828, −5.25402688146794948079464901503, −3.99403219891259908495918356597, −2.88374406410455142618704620052, −2.12782144789846372403009793372, −0.85526618304564913751494492371, 0.24704737836385679077592904813, 1.66459471064338123885467229207, 3.66933362696989145048028768407, 4.158262511772334790760545061546, 5.23395302163544445132876545387, 6.06844597513559303626355842490, 6.71777203555932320399973222845, 7.48881663220829915117300817279, 8.69677310430059740064954050068, 9.53446169585495453490177437421, 10.15236996110709096596892693335, 10.83224141839716217578873938042, 12.02498679778995092295111560172, 12.84214694636383163963542677970, 13.83098501563859294849138918495, 14.61047527198421177118411483604, 15.48650431643691158748680704933, 16.21580708099084910906324452772, 16.73140338573600012563543865453, 17.32445852628715886251630743405, 18.15623208377893577686566774441, 19.08560711135058406810471938707, 19.71080943257989983986851891282, 20.86343393420598142587566690675, 21.87680307252536513399423759469

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