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Properties

Label 1-179-179.7-r1-0-0
Degree $1$
Conductor $179$
Sign $-0.763 + 0.645i$
Analytic cond. $19.2362$
Root an. cond. $19.2362$
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  + (0.969 − 0.244i)2-s + (0.0176 − 0.999i)3-s + (0.880 − 0.474i)4-s + (−0.896 − 0.442i)5-s + (−0.227 − 0.973i)6-s + (−0.158 + 0.987i)7-s + (0.737 − 0.675i)8-s + (−0.999 − 0.0352i)9-s + (−0.977 − 0.210i)10-s + (−0.844 + 0.535i)11-s + (−0.458 − 0.888i)12-s + (−0.994 − 0.105i)13-s + (0.0881 + 0.996i)14-s + (−0.458 + 0.888i)15-s + (0.550 − 0.835i)16-s + (−0.984 + 0.175i)17-s + ⋯
L(s)  = 1  + (0.969 − 0.244i)2-s + (0.0176 − 0.999i)3-s + (0.880 − 0.474i)4-s + (−0.896 − 0.442i)5-s + (−0.227 − 0.973i)6-s + (−0.158 + 0.987i)7-s + (0.737 − 0.675i)8-s + (−0.999 − 0.0352i)9-s + (−0.977 − 0.210i)10-s + (−0.844 + 0.535i)11-s + (−0.458 − 0.888i)12-s + (−0.994 − 0.105i)13-s + (0.0881 + 0.996i)14-s + (−0.458 + 0.888i)15-s + (0.550 − 0.835i)16-s + (−0.984 + 0.175i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.763 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.763 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $-0.763 + 0.645i$
Analytic conductor: \(19.2362\)
Root analytic conductor: \(19.2362\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (1:\ ),\ -0.763 + 0.645i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2480289814 - 0.6771837449i\)
\(L(\frac12)\) \(\approx\) \(-0.2480289814 - 0.6771837449i\)
\(L(1)\) \(\approx\) \(0.9791075155 - 0.6141614346i\)
\(L(1)\) \(\approx\) \(0.9791075155 - 0.6141614346i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (0.969 - 0.244i)T \)
3 \( 1 + (0.0176 - 0.999i)T \)
5 \( 1 + (-0.896 - 0.442i)T \)
7 \( 1 + (-0.158 + 0.987i)T \)
11 \( 1 + (-0.844 + 0.535i)T \)
13 \( 1 + (-0.994 - 0.105i)T \)
17 \( 1 + (-0.984 + 0.175i)T \)
19 \( 1 + (0.713 - 0.700i)T \)
23 \( 1 + (-0.362 - 0.932i)T \)
29 \( 1 + (-0.635 + 0.772i)T \)
31 \( 1 + (-0.925 - 0.378i)T \)
37 \( 1 + (0.635 + 0.772i)T \)
41 \( 1 + (0.825 - 0.564i)T \)
43 \( 1 + (0.607 - 0.794i)T \)
47 \( 1 + (-0.863 - 0.505i)T \)
53 \( 1 + (0.0529 + 0.998i)T \)
59 \( 1 + (-0.261 - 0.965i)T \)
61 \( 1 + (0.489 + 0.871i)T \)
67 \( 1 + (-0.737 - 0.675i)T \)
71 \( 1 + (-0.427 - 0.904i)T \)
73 \( 1 + (-0.295 - 0.955i)T \)
79 \( 1 + (-0.911 - 0.411i)T \)
83 \( 1 + (0.938 + 0.345i)T \)
89 \( 1 + (-0.969 - 0.244i)T \)
97 \( 1 + (-0.662 + 0.749i)T \)
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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

−27.36590925527777686926179842633, −26.52389148356134030247524328385, −26.14517283084599201931924103222, −24.56853295083726912979029185627, −23.61141235258156624360307884758, −22.83831976743128097149414312275, −22.118603581010946422810309452146, −21.112142895123313802297025084, −20.11114870492689431938057631559, −19.51092999198152606931002446401, −17.64825448902052621369922880587, −16.33670817706356813435008741339, −15.97640863387531937046538421176, −14.85694942034087194418776949984, −14.125602359950132251407091792587, −12.96230097990612587264897806072, −11.5027419323395254701146340821, −10.95969195071199741284108344761, −9.80935689930405203241335437027, −7.99593945250241378845772420319, −7.24623804806493670419499097448, −5.74829799961550145334884320221, −4.53932397161298399834323362847, −3.74322722379555410572439913072, −2.74239292652577536830869898619, 0.1663326381152179004834555223, 2.04805370186001245192209063142, 2.95114270744019012135021738710, 4.65210818197703274164401414792, 5.579478000909187199651795379108, 6.945090668734965053359322575821, 7.78534587306156201369118352489, 9.14322772203223333085800749686, 10.919424829820021423499375400894, 11.94083518181172381494929468045, 12.56121080661829764204557637569, 13.242293877118473057771028067125, 14.70777156409143753431372903853, 15.40052520994343632258510738707, 16.47216114946464293066062418735, 18.00053230423380262776838381275, 18.97199380779223808415301332181, 19.90672012685517909355789276937, 20.48742498830150197726936065707, 22.062164893841629933508468302659, 22.64463447320246379546200122321, 23.9600967169676861545434974205, 24.182837873910053128628225606683, 25.15411884391008762318057992829, 26.27391831796598701031190721963

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