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Properties

Label 1-1037-1037.112-r0-0-0
Degree $1$
Conductor $1037$
Sign $0.862 - 0.505i$
Analytic cond. $4.81580$
Root an. cond. $4.81580$
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  + (−0.998 − 0.0523i)2-s + (−0.996 + 0.0784i)3-s + (0.994 + 0.104i)4-s + (−0.688 + 0.725i)5-s + (0.999 − 0.0261i)6-s + (−0.566 + 0.824i)7-s + (−0.987 − 0.156i)8-s + (0.987 − 0.156i)9-s + (0.725 − 0.688i)10-s + (−0.923 − 0.382i)11-s + (−0.999 − 0.0261i)12-s + (0.866 + 0.5i)13-s + (0.608 − 0.793i)14-s + (0.629 − 0.777i)15-s + (0.978 + 0.207i)16-s + ⋯
L(s)  = 1  + (−0.998 − 0.0523i)2-s + (−0.996 + 0.0784i)3-s + (0.994 + 0.104i)4-s + (−0.688 + 0.725i)5-s + (0.999 − 0.0261i)6-s + (−0.566 + 0.824i)7-s + (−0.987 − 0.156i)8-s + (0.987 − 0.156i)9-s + (0.725 − 0.688i)10-s + (−0.923 − 0.382i)11-s + (−0.999 − 0.0261i)12-s + (0.866 + 0.5i)13-s + (0.608 − 0.793i)14-s + (0.629 − 0.777i)15-s + (0.978 + 0.207i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1037 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1037 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1037\)    =    \(17 \cdot 61\)
Sign: $0.862 - 0.505i$
Analytic conductor: \(4.81580\)
Root analytic conductor: \(4.81580\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1037} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1037,\ (0:\ ),\ 0.862 - 0.505i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2470004530 - 0.06706083276i\)
\(L(\frac12)\) \(\approx\) \(0.2470004530 - 0.06706083276i\)
\(L(1)\) \(\approx\) \(0.3687522798 + 0.06607070130i\)
\(L(1)\) \(\approx\) \(0.3687522798 + 0.06607070130i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
61 \( 1 \)
good2 \( 1 + (-0.998 - 0.0523i)T \)
3 \( 1 + (-0.996 + 0.0784i)T \)
5 \( 1 + (-0.688 + 0.725i)T \)
7 \( 1 + (-0.566 + 0.824i)T \)
11 \( 1 + (-0.923 - 0.382i)T \)
13 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (-0.838 - 0.544i)T \)
23 \( 1 + (0.522 + 0.852i)T \)
29 \( 1 + (-0.608 + 0.793i)T \)
31 \( 1 + (0.333 - 0.942i)T \)
37 \( 1 + (-0.649 - 0.760i)T \)
41 \( 1 + (0.0784 - 0.996i)T \)
43 \( 1 + (-0.629 + 0.777i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (0.156 - 0.987i)T \)
59 \( 1 + (-0.0523 + 0.998i)T \)
67 \( 1 + (-0.406 - 0.913i)T \)
71 \( 1 + (-0.999 - 0.0261i)T \)
73 \( 1 + (0.999 + 0.0261i)T \)
79 \( 1 + (0.284 - 0.958i)T \)
83 \( 1 + (-0.0523 + 0.998i)T \)
89 \( 1 + (-0.309 + 0.951i)T \)
97 \( 1 + (0.333 - 0.942i)T \)
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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

−21.31126398807830062840412994894, −20.67545662067805817723454217876, −20.079728031789930152165915629301, −19.03289493119240834334446894165, −18.578359137825569715055967766663, −17.58261944228267770509660022255, −16.92222172080595461740778187324, −16.34734345843444976615452307452, −15.72879811980793533441660080122, −15.05457998742821483923075682595, −13.32282535851652915824590289705, −12.74499098664814059800139743951, −11.99053438451260895446166921465, −11.01015229221313570880779784484, −10.4682105371260751529323992102, −9.803069825832721454387144174490, −8.54601728350489470778162467, −7.909736898901739267830693703677, −7.02712747662318637111206637832, −6.29771425097212034400453185933, −5.28288589132837257825483922753, −4.287038193268642836880969880156, −3.20425216316921207114698850101, −1.64404163828096781815790735325, −0.68323651491672661276327847389, 0.28071766763155849603283873793, 1.82387965406517806981336923027, 2.93169913896629137975732790532, 3.827504583343122640575962979849, 5.30988379926281800205591406168, 6.17611851453107809561907280321, 6.79423006669419753578875330268, 7.65260036716748420064493860749, 8.636881232796431924143890617928, 9.48093447559694357423898818286, 10.48700979451076050249329246150, 11.07531627802898748470526211593, 11.5795924777520127063431262351, 12.4908646994228325407160102680, 13.301941923670083231287400247827, 14.96041166245723695415391291028, 15.51196196303814541738266064592, 16.09277310276087700387236838525, 16.72325634761669083272585363315, 17.91496185146638626599012474962, 18.27466396684358355356506347225, 19.11928551631994571766765633321, 19.38203623437922971543765199190, 20.8981411959712929331595364906, 21.390657142356820802991993036666

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