L-function 1-175-175.17-r0-0-0

• Label: 1-175-175.17-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: $0$

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 + ⋯

Functional equation

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

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:	$\chi_{175} (17, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 175,\ (0:\ ),\ 0.388 - 0.921i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.622262936 - 1.076558827i$
$L(1)$	$\approx$	$1.601662460 - 0.6539822273i$

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	5	$1$
	7	$1$
good	2	$1 + (0.994 - 0.104i)T$
	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	$1 + (-0.669 + 0.743i)T$

Imaginary part of the first few zeros on the critical line

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

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