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Properties

Label 1-179-179.19-r0-0-0
Degree $1$
Conductor $179$
Sign $0.201 - 0.979i$
Analytic cond. $0.831272$
Root an. cond. $0.831272$
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  + (−0.329 + 0.944i)2-s + (0.0881 − 0.996i)3-s + (−0.783 − 0.621i)4-s + (0.662 − 0.749i)5-s + (0.911 + 0.411i)6-s + (0.713 − 0.700i)7-s + (0.844 − 0.535i)8-s + (−0.984 − 0.175i)9-s + (0.489 + 0.871i)10-s + (−0.949 − 0.312i)11-s + (−0.688 + 0.725i)12-s + (−0.863 − 0.505i)13-s + (0.427 + 0.904i)14-s + (−0.688 − 0.725i)15-s + (0.227 + 0.973i)16-s + (−0.635 + 0.772i)17-s + ⋯
L(s)  = 1  + (−0.329 + 0.944i)2-s + (0.0881 − 0.996i)3-s + (−0.783 − 0.621i)4-s + (0.662 − 0.749i)5-s + (0.911 + 0.411i)6-s + (0.713 − 0.700i)7-s + (0.844 − 0.535i)8-s + (−0.984 − 0.175i)9-s + (0.489 + 0.871i)10-s + (−0.949 − 0.312i)11-s + (−0.688 + 0.725i)12-s + (−0.863 − 0.505i)13-s + (0.427 + 0.904i)14-s + (−0.688 − 0.725i)15-s + (0.227 + 0.973i)16-s + (−0.635 + 0.772i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $0.201 - 0.979i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ 0.201 - 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6820427226 - 0.5559826184i\)
\(L(\frac12)\) \(\approx\) \(0.6820427226 - 0.5559826184i\)
\(L(1)\) \(\approx\) \(0.8561878962 - 0.2096301033i\)
\(L(1)\) \(\approx\) \(0.8561878962 - 0.2096301033i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (-0.329 + 0.944i)T \)
3 \( 1 + (0.0881 - 0.996i)T \)
5 \( 1 + (0.662 - 0.749i)T \)
7 \( 1 + (0.713 - 0.700i)T \)
11 \( 1 + (-0.949 - 0.312i)T \)
13 \( 1 + (-0.863 - 0.505i)T \)
17 \( 1 + (-0.635 + 0.772i)T \)
19 \( 1 + (-0.737 + 0.675i)T \)
23 \( 1 + (0.960 - 0.278i)T \)
29 \( 1 + (0.295 - 0.955i)T \)
31 \( 1 + (0.362 - 0.932i)T \)
37 \( 1 + (0.295 + 0.955i)T \)
41 \( 1 + (0.990 + 0.140i)T \)
43 \( 1 + (-0.123 + 0.992i)T \)
47 \( 1 + (0.880 - 0.474i)T \)
53 \( 1 + (-0.261 - 0.965i)T \)
59 \( 1 + (-0.969 - 0.244i)T \)
61 \( 1 + (0.550 - 0.835i)T \)
67 \( 1 + (0.844 + 0.535i)T \)
71 \( 1 + (0.804 - 0.593i)T \)
73 \( 1 + (0.997 + 0.0705i)T \)
79 \( 1 + (-0.520 + 0.854i)T \)
83 \( 1 + (-0.192 + 0.981i)T \)
89 \( 1 + (-0.329 - 0.944i)T \)
97 \( 1 + (-0.458 + 0.888i)T \)
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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

−27.47512125537165287242597494987, −26.797938985688124425435841878264, −26.00463217419677917017035391164, −25.07465112685186215797862032194, −23.34514519430896652770241321083, −22.27364787978625631256276517966, −21.51206976845770361519112041828, −21.13230748199957629357940463341, −19.9674537919109686928090174227, −18.837275354520289317207530433114, −17.84646916921213771412267262063, −17.2018184017115750661636697502, −15.6909935399643459381567452501, −14.654355878746800088126308713528, −13.794525185716172820881393898426, −12.41090026946624474580327724190, −11.112107716234725486552593922253, −10.64662012304194639518676717059, −9.45193770588287676426920924904, −8.807154156182553642659002900526, −7.28812913833154953590109760102, −5.32166385028086903569794814608, −4.57414596615327127855608392523, −2.83151407198072358904571700482, −2.27222358992510117081245637644, 0.77980912552717900600268243713, 2.16618920039578552088798314090, 4.54114535872577114897050660396, 5.581351312964721231692368989577, 6.60455300913404373231978606144, 7.96818440292502600185318598069, 8.28388773294692139374697914421, 9.764544109569965588135040633834, 10.9199181597699589212003941664, 12.72780674881853854809097215366, 13.29666317005551753032577051037, 14.25677080681896432432090237577, 15.25064697628750422211238536727, 16.90432515210227493535538205417, 17.20299216503455137622258096067, 18.11737642937482221417779770365, 19.15628076035078409264085976065, 20.19832010333813222598388818745, 21.29216720795485427720337457128, 22.80843952450390290930493758329, 23.73277318892828701275550821603, 24.38066618688482781105941167758, 24.99069876709569726799109071580, 26.03314151883667594820243678525, 26.9002530156064966926235202573

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