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Properties

Label 1-1033-1033.1019-r0-0-0
Degree $1$
Conductor $1033$
Sign $-0.914 - 0.403i$
Analytic cond. $4.79723$
Root an. cond. $4.79723$
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  + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.866 − 0.5i)5-s − 6-s − 7-s + 8-s + (−0.5 − 0.866i)9-s + (−0.866 − 0.5i)10-s + (0.866 − 0.5i)11-s + (0.5 + 0.866i)12-s + (0.866 − 0.5i)13-s + (0.5 + 0.866i)14-s − i·15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.866 − 0.5i)5-s − 6-s − 7-s + 8-s + (−0.5 − 0.866i)9-s + (−0.866 − 0.5i)10-s + (0.866 − 0.5i)11-s + (0.5 + 0.866i)12-s + (0.866 − 0.5i)13-s + (0.5 + 0.866i)14-s − i·15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.914 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.914 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1033\)
Sign: $-0.914 - 0.403i$
Analytic conductor: \(4.79723\)
Root analytic conductor: \(4.79723\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1033} (1019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1033,\ (0:\ ),\ -0.914 - 0.403i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3181760950 - 1.509256294i\)
\(L(\frac12)\) \(\approx\) \(0.3181760950 - 1.509256294i\)
\(L(1)\) \(\approx\) \(0.7375120192 - 0.8206839584i\)
\(L(1)\) \(\approx\) \(0.7375120192 - 0.8206839584i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1033 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 - T \)
11 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (0.866 - 0.5i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + iT \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 - T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.866 - 0.5i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−21.859780879377504180182482713988, −21.36432035688217974037529952900, −20.18098252849522037593955201442, −19.38797394416064209340836824932, −18.90986136802762486350927873228, −17.81155758497854284038099287818, −17.03282691435611343869367293880, −16.49433162754134042222913914099, −15.61546631706604494342494098266, −14.93462550370913743322374150017, −14.231552735421468508659909459699, −13.59997894445880595430180848395, −12.69740141704057168852389055352, −11.04002678869813013894139011647, −10.40558503954163632441990420329, −9.7056206901459000940496185350, −9.02537122113143433421640560954, −8.50853018240811770411779550399, −7.01766056691866734650695463992, −6.51443353618068797232172447590, −5.69658978657981187247647901675, −4.6029947424910656758281948304, −3.691442081600699808023728886284, −2.591557531630373203672878981878, −1.36822016682807124362881877395, 0.85503960189500045954103087073, 1.39613336160341031204017254501, 2.70791131282136222746372576120, 3.205320297510164309186077013604, 4.33778253445115824566202632487, 5.9282368506284547238918680242, 6.41447403930166793567338389868, 7.69515732919429670669691713085, 8.43945695472483755128287465438, 9.40558444377150940741990137947, 9.58093409367757724164930440761, 10.829352189238000886158409239226, 11.84479659095723429134724940068, 12.54059525423123569771840861814, 13.308230168127287094955770907865, 13.63772752160711650405278856627, 14.60090132919020987004160902318, 16.08241026003951525577125031277, 16.79254448672952875079380506661, 17.50384263510165637055549695667, 18.33984331241843274693734939335, 18.96274825578809412019196729360, 19.60597861569733929269087858996, 20.37880834592014763872213529917, 21.0039566239537020590748664651

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