Properties

Label 1-175-175.6-r1-0-0
Degree $1$
Conductor $175$
Sign $0.968 - 0.248i$
Analytic cond. $18.8063$
Root an. cond. $18.8063$
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  + (−0.809 + 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 − 0.951i)4-s + (−0.309 − 0.951i)6-s + (0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + (−0.809 + 0.587i)11-s + (0.809 + 0.587i)12-s + (0.809 + 0.587i)13-s + (−0.809 − 0.587i)16-s + (−0.309 − 0.951i)17-s + 18-s + (−0.309 − 0.951i)19-s + (0.309 − 0.951i)22-s + (−0.809 + 0.587i)23-s − 24-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 − 0.951i)4-s + (−0.309 − 0.951i)6-s + (0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + (−0.809 + 0.587i)11-s + (0.809 + 0.587i)12-s + (0.809 + 0.587i)13-s + (−0.809 − 0.587i)16-s + (−0.309 − 0.951i)17-s + 18-s + (−0.309 − 0.951i)19-s + (0.309 − 0.951i)22-s + (−0.809 + 0.587i)23-s − 24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.968 - 0.248i$
Analytic conductor: \(18.8063\)
Root analytic conductor: \(18.8063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (1:\ ),\ 0.968 - 0.248i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6013196611 - 0.07596433904i\)
\(L(\frac12)\) \(\approx\) \(0.6013196611 - 0.07596433904i\)
\(L(1)\) \(\approx\) \(0.5458606123 + 0.2161215053i\)
\(L(1)\) \(\approx\) \(0.5458606123 + 0.2161215053i\)

Euler product

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

−27.47371758475349082898411555084, −26.19909118147430286936917616274, −25.51833638778556373047660958003, −24.460734199939552102957848582427, −23.47866201662362965844715517259, −22.37285708809800172052132073629, −21.25734460997500698228001072468, −20.24093813964151158137023397654, −19.27656947387008834560863923381, −18.42809858510330884812112898947, −17.81031057040094621983951774807, −16.71448316245106631316833068247, −15.79371230266010215111053568762, −14.02960470195707666734812222572, −12.89466777408276159454515501399, −12.30160338220385269641797779503, −10.942658913647661343094775430832, −10.41667841069620331955202682931, −8.54784914988733120954430510635, −8.11506513423923225930048512611, −6.78225246505186888490054955960, −5.6502003377106743855066998425, −3.62148682494673382611530757629, −2.28918857333759421327201386378, −1.05143139152508033627291656638, 0.34899518859678038294285593213, 2.37959203083080500561405437902, 4.281243365962627288819669576853, 5.37597516845343868774678213885, 6.47995688376269356703008841043, 7.776051344930300001530111404406, 9.035373013617821036333935326459, 9.75915430821709164536638475644, 10.85380340059986373974750292806, 11.6359289995614848869395674176, 13.50432432191175427684286313544, 14.67413218994454365415213761123, 15.79098910970818915661058599448, 16.065003596805141610022273875348, 17.440056161724163967209944660178, 18.02379216404834484630746855833, 19.29911844016297585759646788151, 20.43954899334239872914952147694, 21.14713300065436512266469116709, 22.55453425784415878019103567053, 23.376005051538622711957401938, 24.30060535440157885904251321788, 25.77831848903295916146222595133, 26.09414145262795180852671484152, 27.12850971287618650385418831174

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