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Properties

Label 1-1045-1045.278-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.988 + 0.151i$
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  + (−0.743 + 0.669i)2-s + (−0.994 + 0.104i)3-s + (0.104 − 0.994i)4-s + (0.669 − 0.743i)6-s + (−0.587 + 0.809i)7-s + (0.587 + 0.809i)8-s + (0.978 − 0.207i)9-s + i·12-s + (0.207 + 0.978i)13-s + (−0.104 − 0.994i)14-s + (−0.978 − 0.207i)16-s + (−0.207 + 0.978i)17-s + (−0.587 + 0.809i)18-s + (0.5 − 0.866i)21-s + (0.866 − 0.5i)23-s + (−0.669 − 0.743i)24-s + ⋯
L(s)  = 1  + (−0.743 + 0.669i)2-s + (−0.994 + 0.104i)3-s + (0.104 − 0.994i)4-s + (0.669 − 0.743i)6-s + (−0.587 + 0.809i)7-s + (0.587 + 0.809i)8-s + (0.978 − 0.207i)9-s + i·12-s + (0.207 + 0.978i)13-s + (−0.104 − 0.994i)14-s + (−0.978 − 0.207i)16-s + (−0.207 + 0.978i)17-s + (−0.587 + 0.809i)18-s + (0.5 − 0.866i)21-s + (0.866 − 0.5i)23-s + (−0.669 − 0.743i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.988 + 0.151i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (278, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.988 + 0.151i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03572159556 + 0.4702280411i\)
\(L(\frac12)\) \(\approx\) \(0.03572159556 + 0.4702280411i\)
\(L(1)\) \(\approx\) \(0.4306834161 + 0.2742648809i\)
\(L(1)\) \(\approx\) \(0.4306834161 + 0.2742648809i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.743 + 0.669i)T \)
3 \( 1 + (-0.994 + 0.104i)T \)
7 \( 1 + (-0.587 + 0.809i)T \)
13 \( 1 + (0.207 + 0.978i)T \)
17 \( 1 + (-0.207 + 0.978i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (-0.104 + 0.994i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (0.587 - 0.809i)T \)
41 \( 1 + (0.104 + 0.994i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (0.406 - 0.913i)T \)
53 \( 1 + (0.207 + 0.978i)T \)
59 \( 1 + (0.913 - 0.406i)T \)
61 \( 1 + (0.669 - 0.743i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (0.978 + 0.207i)T \)
73 \( 1 + (0.406 + 0.913i)T \)
79 \( 1 + (0.669 + 0.743i)T \)
83 \( 1 + (-0.951 + 0.309i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.743 + 0.669i)T \)
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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

−20.91782568746340189166110021169, −20.47572275362205454756745515084, −19.47590392574509332219907479945, −18.83303692871768986387313361872, −17.97653533004209047979827049998, −17.38225074350278657777107618288, −16.701302024298379321651632461029, −16.038963593903117827807303690124, −15.21976502861780999082553215761, −13.4792897229918832303935062404, −13.19792675002323067328371426382, −12.28278397055161417221246069659, −11.36421157596318682562764936831, −10.89538211690862105056503045013, −9.931182218727574439670038926331, −9.53563180307287654482993030331, −8.15705021239363516439553545801, −7.33565947192711955455429410017, −6.70075885706680910215973557581, −5.58233565930118310014237027531, −4.46885738723535212642234991436, −3.57317371121114031262773378668, −2.528769833752718018011375891222, −1.13871858870082594134971932060, −0.37449060884446454406212215451, 1.18203011020992720603015709291, 2.22023464703618140024181523728, 3.823345266263208373730388847070, 4.99532585114729657556251671799, 5.64764289592876849002439838876, 6.65900256364445832664053825395, 6.85927642621445294691097922237, 8.31040858214072908998056526571, 9.06400438853410907881178907651, 9.79036407584743621636054233594, 10.72413808266566224107573216854, 11.326073405185637120070626885022, 12.34952559048353827839641378820, 13.07511242743595503665558791227, 14.334771442719910647688614948717, 15.157799130287897985178074149755, 15.85010845693072143197826567766, 16.577376323197659457315876447532, 17.03031241754490898798731306339, 18.094351051706517202191150499160, 18.55366717347511709560390792297, 19.25267835255640664939298956202, 20.1245187961156002493682406025, 21.485743183989127976605649763452, 21.8202091431667160742725279036

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