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Properties

Label 1-2475-2475.1913-r1-0-0
Degree $1$
Conductor $2475$
Sign $0.355 - 0.934i$
Analytic cond. $265.975$
Root an. cond. $265.975$
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  + (−0.207 + 0.978i)2-s + (−0.913 − 0.406i)4-s + (−0.866 − 0.5i)7-s + (0.587 − 0.809i)8-s + (0.207 + 0.978i)13-s + (0.669 − 0.743i)14-s + (0.669 + 0.743i)16-s + (−0.587 + 0.809i)17-s + (−0.809 − 0.587i)19-s + (−0.743 − 0.669i)23-s − 26-s + (0.587 + 0.809i)28-s + (0.104 + 0.994i)29-s + (−0.104 + 0.994i)31-s + (−0.866 + 0.5i)32-s + ⋯
L(s)  = 1  + (−0.207 + 0.978i)2-s + (−0.913 − 0.406i)4-s + (−0.866 − 0.5i)7-s + (0.587 − 0.809i)8-s + (0.207 + 0.978i)13-s + (0.669 − 0.743i)14-s + (0.669 + 0.743i)16-s + (−0.587 + 0.809i)17-s + (−0.809 − 0.587i)19-s + (−0.743 − 0.669i)23-s − 26-s + (0.587 + 0.809i)28-s + (0.104 + 0.994i)29-s + (−0.104 + 0.994i)31-s + (−0.866 + 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.355 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.355 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.355 - 0.934i$
Analytic conductor: \(265.975\)
Root analytic conductor: \(265.975\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (1913, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2475,\ (1:\ ),\ 0.355 - 0.934i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03087842828 + 0.02130160083i\)
\(L(\frac12)\) \(\approx\) \(0.03087842828 + 0.02130160083i\)
\(L(1)\) \(\approx\) \(0.5834689296 + 0.3233447661i\)
\(L(1)\) \(\approx\) \(0.5834689296 + 0.3233447661i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.207 + 0.978i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (0.207 + 0.978i)T \)
17 \( 1 + (-0.587 + 0.809i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.743 - 0.669i)T \)
29 \( 1 + (0.104 + 0.994i)T \)
31 \( 1 + (-0.104 + 0.994i)T \)
37 \( 1 + (-0.951 - 0.309i)T \)
41 \( 1 + (-0.978 + 0.207i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (0.994 - 0.104i)T \)
53 \( 1 + (0.587 + 0.809i)T \)
59 \( 1 + (-0.978 + 0.207i)T \)
61 \( 1 + (0.978 + 0.207i)T \)
67 \( 1 + (0.994 + 0.104i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (0.951 - 0.309i)T \)
79 \( 1 + (-0.104 - 0.994i)T \)
83 \( 1 + (0.406 + 0.913i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.994 + 0.104i)T \)
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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

−19.39642218287822826714704484054, −18.81379069227845513482122153706, −18.24853452944914427460162591204, −17.40059819354008844992489307774, −16.845391983560585198020510969557, −15.72263787038448134854108329763, −15.35890851948973110479079161044, −14.15204595055110698020270225790, −13.47998158985498614401866229848, −12.86600131176318108575891164847, −12.19342391095905378468617452364, −11.54993464481291765217811757228, −10.6822526220053737448477378428, −9.96407059083726209227202971997, −9.45026498198159363809620728346, −8.55979632274837259199103118507, −7.96982537616610600278846606951, −6.923727051355423743819943632706, −5.88131724541454362370984752101, −5.26402701202707438738689613300, −4.091586319950711030485102320673, −3.51791289688358875893057576864, −2.56380963832335032057684020982, −1.99457024497647663106839549820, −0.63197283937866568705780084091, 0.01084486772511762319100721968, 1.13426470585481034930970024577, 2.23866071550910016399233893374, 3.60518826525276361905633788516, 4.176017777966464761232755928341, 4.99736019708866452264561101978, 6.07537791237072013618025313542, 6.67869813124254146759848965006, 7.07147119242043087523869111604, 8.20741641971119014504553228588, 8.873639053708962574816787654173, 9.4208097278993448505581886802, 10.50528150716258636985021630594, 10.76416863942663605331222754412, 12.25229028727502282124257102218, 12.78704666832608129310618047103, 13.69230792283495799466257247247, 14.09190110068646467748941683065, 15.012238145589678946290751232752, 15.71188388519959380439961466334, 16.35535030079981711795840799621, 16.89304536442434495208475133974, 17.59367784939103207393246760480, 18.35737307927272166762037808174, 19.16489763058214203745559021850

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