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Properties

Label 1-705-705.563-r1-0-0
Degree $1$
Conductor $705$
Sign $0.850 - 0.525i$
Analytic cond. $75.7627$
Root an. cond. $75.7627$
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 − 4-s − i·7-s − i·8-s + 11-s − i·13-s + 14-s + 16-s + i·17-s + 19-s + i·22-s + i·23-s + 26-s + i·28-s − 29-s + ⋯
L(s)  = 1  + i·2-s − 4-s − i·7-s − i·8-s + 11-s − i·13-s + 14-s + 16-s + i·17-s + 19-s + i·22-s + i·23-s + 26-s + i·28-s − 29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 705 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(705\)    =    \(3 \cdot 5 \cdot 47\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(75.7627\)
Root analytic conductor: \(75.7627\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{705} (563, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 705,\ (1:\ ),\ 0.850 - 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.495174360 - 0.4247477027i\)
\(L(\frac12)\) \(\approx\) \(1.495174360 - 0.4247477027i\)
\(L(1)\) \(\approx\) \(0.9771146479 + 0.2306654787i\)
\(L(1)\) \(\approx\) \(0.9771146479 + 0.2306654787i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
47 \( 1 \)
good2 \( 1 \)
7 \( 1 \)
11 \( 1 - T \)
13 \( 1 \)
17 \( 1 \)
19 \( 1 - iT \)
23 \( 1 - iT \)
29 \( 1 \)
31 \( 1 \)
37 \( 1 + T \)
41 \( 1 \)
43 \( 1 - iT \)
53 \( 1 \)
59 \( 1 + T \)
61 \( 1 + iT \)
67 \( 1 \)
71 \( 1 + T \)
73 \( 1 \)
79 \( 1 \)
83 \( 1 + iT \)
89 \( 1 + iT \)
97 \( 1 \)
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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

−22.2606155344756636490133531148, −21.7840825150092647680726676838, −20.78990915911941104771395272560, −20.165746554331653060229287815838, −19.18988633327304883456785626960, −18.56711610571793586469455418694, −17.96193779751040969845356338220, −16.834022340849427707084576570239, −16.07032052974774150002280440327, −14.71246372586721138864429585255, −14.25690495085433710938200289149, −13.24087961084227114015924746539, −12.26220907165881731209740972413, −11.66926037551313897945119281253, −11.084014597872016877329333726850, −9.6100668205564168446561191373, −9.31196357342739859781583872005, −8.448093940701121275806021123721, −7.168358100671031598483064174572, −5.98749317691103745642893414712, −5.00782032474336184364786872058, −4.08832837234688769181287909436, −3.01445807718678950301832060011, −2.099772274094888014373002523190, −1.0646349749383361210512219882, 0.42744687676741128103384335276, 1.48243738582670686372403025506, 3.54751979240133279086027210374, 3.955583785904851645502121613, 5.29904121678265422422669387928, 5.99512582620525619342364746240, 7.18969797908644738694606094943, 7.588117192068119962152599557917, 8.70633557013758798383723085259, 9.59143194169152373455026250514, 10.40676031737036445877319529486, 11.468635495774620101000718721763, 12.75701634385745070096870361187, 13.3301351489068793252276093197, 14.32877065860241820400250818227, 14.83421370854445717842033333911, 15.89357999156500694075192995003, 16.60265852864845893450208543668, 17.48120251673838022767519111611, 17.809574478405499128788000696255, 19.11921730373225099260056108613, 19.79027162921732071324485706068, 20.66824204419052790684564298726, 21.92915703104410746234621483640, 22.45718161795086335419191071606

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