Properties

Label 1-175-175.103-r0-0-0
Degree $1$
Conductor $175$
Sign $0.388 + 0.921i$
Analytic cond. $0.812696$
Root an. cond. $0.812696$
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  + (0.994 + 0.104i)2-s + (−0.207 + 0.978i)3-s + (0.978 + 0.207i)4-s + (−0.309 + 0.951i)6-s + (0.951 + 0.309i)8-s + (−0.913 − 0.406i)9-s + (0.913 − 0.406i)11-s + (−0.406 + 0.913i)12-s + (0.587 + 0.809i)13-s + (0.913 + 0.406i)16-s + (−0.743 + 0.669i)17-s + (−0.866 − 0.5i)18-s + (−0.978 + 0.207i)19-s + (0.951 − 0.309i)22-s + (−0.994 − 0.104i)23-s + (−0.5 + 0.866i)24-s + ⋯
L(s)  = 1  + (0.994 + 0.104i)2-s + (−0.207 + 0.978i)3-s + (0.978 + 0.207i)4-s + (−0.309 + 0.951i)6-s + (0.951 + 0.309i)8-s + (−0.913 − 0.406i)9-s + (0.913 − 0.406i)11-s + (−0.406 + 0.913i)12-s + (0.587 + 0.809i)13-s + (0.913 + 0.406i)16-s + (−0.743 + 0.669i)17-s + (−0.866 − 0.5i)18-s + (−0.978 + 0.207i)19-s + (0.951 − 0.309i)22-s + (−0.994 − 0.104i)23-s + (−0.5 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.388 + 0.921i$
Analytic conductor: \(0.812696\)
Root analytic conductor: \(0.812696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (0:\ ),\ 0.388 + 0.921i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.622262936 + 1.076558827i\)
\(L(\frac12)\) \(\approx\) \(1.622262936 + 1.076558827i\)
\(L(1)\) \(\approx\) \(1.601662460 + 0.6539822273i\)
\(L(1)\) \(\approx\) \(1.601662460 + 0.6539822273i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.994 + 0.104i)T \)
3 \( 1 + (-0.207 + 0.978i)T \)
11 \( 1 + (0.913 - 0.406i)T \)
13 \( 1 + (0.587 + 0.809i)T \)
17 \( 1 + (-0.743 + 0.669i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (-0.994 - 0.104i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (0.406 - 0.913i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 + iT \)
47 \( 1 + (0.743 + 0.669i)T \)
53 \( 1 + (0.207 - 0.978i)T \)
59 \( 1 + (-0.104 - 0.994i)T \)
61 \( 1 + (0.104 - 0.994i)T \)
67 \( 1 + (0.743 - 0.669i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.406 - 0.913i)T \)
79 \( 1 + (-0.669 + 0.743i)T \)
83 \( 1 + (-0.951 - 0.309i)T \)
89 \( 1 + (-0.104 + 0.994i)T \)
97 \( 1 + (-0.951 + 0.309i)T \)
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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

−27.57879128872932669200752536475, −25.71455198325650495708558404887, −25.18027776537632695594293916608, −24.233923453539475173180816751069, −23.40104561021293856275456033584, −22.574740555837861093888402476149, −21.79091450832429806021053302686, −20.214131136231375976169264647105, −19.87852429752722903903150996182, −18.54022270004292536920399830291, −17.49451612741983702981751359072, −16.39138637149834911057030969706, −15.16440745761473664792411197536, −14.15567163137230452508387368356, −13.25220945427561624782425208505, −12.41743828273156903158748381833, −11.51929343600920080356358040888, −10.53216853848234194297556380271, −8.730558833499094589602342491991, −7.35105609223963695918850271325, −6.50169915624328710377262245649, −5.51214757196895842820335794515, −4.11976989428970752676873095570, −2.67592315325194079098662624161, −1.43642388269789115363961812586, 2.13336330704012186958834216368, 3.88387946647996014683402750953, 4.21520453607459295203339616932, 5.83831038912194003797033020766, 6.473559138889382535053967595726, 8.25284061840417407082857916319, 9.463849540677412573358013156082, 10.90005366707380983605721963775, 11.44447525004338929281851780551, 12.681400094339154228074769292148, 13.9906726458829724062442998849, 14.708991889890398521364389528600, 15.73704215752963463604155242110, 16.558723918612264673982276856581, 17.38909896280172437875288887897, 19.21828643428559335371413230914, 20.17953144806403465843382806772, 21.20437542279881815039049465405, 21.83268988252082322401633078208, 22.65555740588279604534685325072, 23.62424961427545982395771168502, 24.54766949495987636823579268791, 25.78877614689738613762496553431, 26.43981851684892310579990698610, 27.77332734851394612391270221806

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