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Properties

Label 1-1045-1045.417-r1-0-0
Degree $1$
Conductor $1045$
Sign $-0.850 - 0.525i$
Analytic cond. $112.300$
Root an. cond. $112.300$
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  + i·2-s + i·3-s − 4-s − 6-s − i·7-s − i·8-s − 9-s − i·12-s − i·13-s + 14-s + 16-s − i·17-s − i·18-s + 21-s − i·23-s + 24-s + ⋯
L(s)  = 1  + i·2-s + i·3-s − 4-s − 6-s − i·7-s − i·8-s − 9-s − i·12-s − i·13-s + 14-s + 16-s − i·17-s − i·18-s + 21-s − i·23-s + 24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.850 - 0.525i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ -0.850 - 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08618615524 + 0.3033879376i\)
\(L(\frac12)\) \(\approx\) \(-0.08618615524 + 0.3033879376i\)
\(L(1)\) \(\approx\) \(0.6613525516 + 0.4087383554i\)
\(L(1)\) \(\approx\) \(0.6613525516 + 0.4087383554i\)

Euler product

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

−20.84914881432352387961734757177, −19.97081370200061708782074842028, −19.09855926942654611959922282964, −18.88289874252776414767936484258, −18.00400609801402791389796590598, −17.341541466504074546161609185627, −16.44135455258174925121068411722, −15.02152864894046874115843644613, −14.459620537016858369441286420156, −13.407854061382781485787824887896, −12.94143230205821033606709371140, −11.98267924050323664249643677934, −11.63008886864186032925074226156, −10.73306505060683261794589835531, −9.499539338525112713939686397935, −8.86978352701435215817134723218, −8.13666498576018543069313635498, −7.08994386754290259276632866646, −5.93019460132012839964409315581, −5.33095184669005947341470578947, −4.00532956450267960053830033603, −3.05944673299176247725057133799, −1.95949988121538654046342061260, −1.60855084693943211680781341038, −0.08370402013394201195611241920, 0.81987859932925604278075656713, 2.84535770606721246302611092890, 3.81888768068722673082611854080, 4.52685577250632722474432662514, 5.35823916746558109031053442824, 6.16258262070617627231157381605, 7.32699992978469771905430394320, 7.89953373447294290319579148347, 8.999860383502215931236175702211, 9.61856686813930107902944530222, 10.50364388431273847196158224412, 11.12286821296966415828372870200, 12.55218852211099252211054953167, 13.34033906609519775159858542816, 14.32604746301232828719538313113, 14.66781935167394634798916784370, 15.717570274442580404638471570978, 16.27047981820202343507485790627, 16.90010516232562469085011823011, 17.66754080553940393621668042132, 18.38354076838746714388005799329, 19.613833017490878490580726542651, 20.33848676068962295944799147355, 21.02871815548059020420452911873, 22.06737056595351825533794427088

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