Properties

Label 1-1015-1015.978-r1-0-0
Degree $1$
Conductor $1015$
Sign $-0.808 + 0.587i$
Analytic cond. $109.076$
Root an. cond. $109.076$
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  + (0.988 + 0.149i)2-s + (0.733 + 0.680i)3-s + (0.955 + 0.294i)4-s + (0.623 + 0.781i)6-s + (0.900 + 0.433i)8-s + (0.0747 + 0.997i)9-s + (−0.997 − 0.0747i)11-s + (0.5 + 0.866i)12-s + (−0.433 − 0.900i)13-s + (0.826 + 0.563i)16-s + (−0.5 + 0.866i)17-s + (−0.0747 + 0.997i)18-s + (0.680 + 0.733i)19-s + (−0.974 − 0.222i)22-s + (0.930 + 0.365i)23-s + (0.365 + 0.930i)24-s + ⋯
L(s)  = 1  + (0.988 + 0.149i)2-s + (0.733 + 0.680i)3-s + (0.955 + 0.294i)4-s + (0.623 + 0.781i)6-s + (0.900 + 0.433i)8-s + (0.0747 + 0.997i)9-s + (−0.997 − 0.0747i)11-s + (0.5 + 0.866i)12-s + (−0.433 − 0.900i)13-s + (0.826 + 0.563i)16-s + (−0.5 + 0.866i)17-s + (−0.0747 + 0.997i)18-s + (0.680 + 0.733i)19-s + (−0.974 − 0.222i)22-s + (0.930 + 0.365i)23-s + (0.365 + 0.930i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.808 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.808 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1015\)    =    \(5 \cdot 7 \cdot 29\)
Sign: $-0.808 + 0.587i$
Analytic conductor: \(109.076\)
Root analytic conductor: \(109.076\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1015} (978, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1015,\ (1:\ ),\ -0.808 + 0.587i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.365934364 + 4.203458310i\)
\(L(\frac12)\) \(\approx\) \(1.365934364 + 4.203458310i\)
\(L(1)\) \(\approx\) \(1.956668300 + 1.125718147i\)
\(L(1)\) \(\approx\) \(1.956668300 + 1.125718147i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.988 + 0.149i)T \)
3 \( 1 + (0.733 + 0.680i)T \)
11 \( 1 + (-0.997 - 0.0747i)T \)
13 \( 1 + (-0.433 - 0.900i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.680 + 0.733i)T \)
23 \( 1 + (0.930 + 0.365i)T \)
31 \( 1 + (-0.930 + 0.365i)T \)
37 \( 1 + (0.0747 + 0.997i)T \)
41 \( 1 - iT \)
43 \( 1 + (0.623 + 0.781i)T \)
47 \( 1 + (-0.826 - 0.563i)T \)
53 \( 1 + (-0.930 + 0.365i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.294 + 0.955i)T \)
67 \( 1 + (0.563 + 0.826i)T \)
71 \( 1 + (0.900 - 0.433i)T \)
73 \( 1 + (-0.988 + 0.149i)T \)
79 \( 1 + (-0.997 + 0.0747i)T \)
83 \( 1 + (-0.974 + 0.222i)T \)
89 \( 1 + (-0.149 + 0.988i)T \)
97 \( 1 + (0.222 + 0.974i)T \)
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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

−21.07348645265238005728157428115, −20.37113704172579695536681309003, −19.77710488618810845421996916641, −18.881930636090326765171962145682, −18.275971008604405699438153232763, −17.17342629104159611364801914591, −16.019409125213178155744163047173, −15.494953424837166927450922372544, −14.45094379611724242542052060825, −14.05250099121985524127326657631, −13.02889227018181272278219857536, −12.77562357247109603062547897817, −11.61100525451911341040588947972, −11.058945920822772965685423101641, −9.75243268109207089555520255282, −9.033119784852238743227390037336, −7.735442593235601566246411032408, −7.17376523308185568413815469885, −6.43511703256321820456260794138, −5.234241975045867365204169191, −4.50029368413159488340524732569, −3.295249338893963829369709769740, −2.580215641021834708286946490009, −1.82682121460592866761027617776, −0.49463717174830231747706294215, 1.59347990573046820887075344619, 2.72880192425219561048485647340, 3.27977022354423218782909479708, 4.26240591064243914712871830400, 5.18568570911895720083825032962, 5.73569915002637998252832962325, 7.14777446694016284559057595981, 7.85230406515757885111702225410, 8.59488159336272833495591581117, 9.88260294840755160081068019918, 10.5634840296872422279783974613, 11.26173458448371233953301061238, 12.552931881628442265104912322677, 13.08999491310455603050625795792, 13.865538810155261322310831265878, 14.76673603759583220263849389458, 15.265234281317079839877159149829, 15.93989291490471305611131360945, 16.70965871537866984786738028377, 17.664205577062599179708219544716, 18.87455870944017658968830275476, 19.73968208155123516695433678127, 20.396717118799764943202232693068, 21.010608330786024403257706014623, 21.71517646525571876562893421530

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