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Properties

Label 1-1045-1045.262-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.995 + 0.0948i$
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  + (−0.970 + 0.241i)2-s + (−0.999 + 0.0348i)3-s + (0.882 − 0.469i)4-s + (0.961 − 0.275i)6-s + (0.743 + 0.669i)7-s + (−0.743 + 0.669i)8-s + (0.997 − 0.0697i)9-s + (−0.866 + 0.5i)12-s + (−0.829 + 0.559i)13-s + (−0.882 − 0.469i)14-s + (0.559 − 0.829i)16-s + (0.0697 − 0.997i)17-s + (−0.951 + 0.309i)18-s + (−0.766 − 0.642i)21-s + (0.984 − 0.173i)23-s + (0.719 − 0.694i)24-s + ⋯
L(s)  = 1  + (−0.970 + 0.241i)2-s + (−0.999 + 0.0348i)3-s + (0.882 − 0.469i)4-s + (0.961 − 0.275i)6-s + (0.743 + 0.669i)7-s + (−0.743 + 0.669i)8-s + (0.997 − 0.0697i)9-s + (−0.866 + 0.5i)12-s + (−0.829 + 0.559i)13-s + (−0.882 − 0.469i)14-s + (0.559 − 0.829i)16-s + (0.0697 − 0.997i)17-s + (−0.951 + 0.309i)18-s + (−0.766 − 0.642i)21-s + (0.984 − 0.173i)23-s + (0.719 − 0.694i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.995 + 0.0948i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.995 + 0.0948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7066978446 + 0.03357732068i\)
\(L(\frac12)\) \(\approx\) \(0.7066978446 + 0.03357732068i\)
\(L(1)\) \(\approx\) \(0.5875827777 + 0.06621041400i\)
\(L(1)\) \(\approx\) \(0.5875827777 + 0.06621041400i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.970 + 0.241i)T \)
3 \( 1 + (-0.999 + 0.0348i)T \)
7 \( 1 + (0.743 + 0.669i)T \)
13 \( 1 + (-0.829 + 0.559i)T \)
17 \( 1 + (0.0697 - 0.997i)T \)
23 \( 1 + (0.984 - 0.173i)T \)
29 \( 1 + (0.848 + 0.529i)T \)
31 \( 1 + (0.104 - 0.994i)T \)
37 \( 1 + (0.951 - 0.309i)T \)
41 \( 1 + (-0.0348 - 0.999i)T \)
43 \( 1 + (-0.984 - 0.173i)T \)
47 \( 1 + (0.927 - 0.374i)T \)
53 \( 1 + (0.898 + 0.438i)T \)
59 \( 1 + (-0.374 + 0.927i)T \)
61 \( 1 + (-0.719 - 0.694i)T \)
67 \( 1 + (-0.642 - 0.766i)T \)
71 \( 1 + (-0.438 - 0.898i)T \)
73 \( 1 + (-0.788 + 0.615i)T \)
79 \( 1 + (0.961 + 0.275i)T \)
83 \( 1 + (0.406 - 0.913i)T \)
89 \( 1 + (-0.939 - 0.342i)T \)
97 \( 1 + (-0.970 + 0.241i)T \)
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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

−21.470882105785080872960995425530, −20.76265901540113613840355785677, −19.781207601899049370315749491901, −19.20675686880800403646291362902, −18.11961534578857234850733603221, −17.65489829693416074634439129692, −16.98664891432320023600872371853, −16.50947480920601987176655176443, −15.380910051308500559742919397331, −14.77615947486676221331391543328, −13.35341822813919202366768262651, −12.53432415016496627055095900708, −11.777508359272089211485670939379, −11.02599516724480339461834941805, −10.3691469205389838705384804458, −9.81167636404588963515237689521, −8.53589304253529037885697830094, −7.733087418780387170427278820272, −7.02861352031848510177711713529, −6.1705174595890472080910256188, −5.080813061631343135220584591529, −4.18390497158592924029069931147, −2.905716954523259058267787222515, −1.60994703165558730316011058872, −0.84738162118755252858128904310, 0.68041051302162602574566933678, 1.81985710619416468253056287024, 2.75287670055085393937114971672, 4.54613250933854145248719866543, 5.23726253318968614976090058906, 6.068878097987956928883672308987, 7.05713544372103880873034708030, 7.59804892900918078358309992594, 8.81080810901248342816155652652, 9.45103793463022715821644928337, 10.38001158378844194454519463739, 11.165636504881889565432962106527, 11.848533255739673015410304028384, 12.34540580611241487484902426842, 13.77607237062220258888405335826, 14.877332010744729784064557487841, 15.38924776122269743497966318033, 16.38100486304064873731646069172, 16.916789883348633545492060993169, 17.65349128682250724007986580876, 18.40714752006268677103916115672, 18.810377978218729526351434148, 19.852623511063326310780591755613, 20.84913112861915136768985758214, 21.461114076548775366560509656535

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