Properties

Label 1-19e2-361.106-r0-0-0
Degree $1$
Conductor $361$
Sign $-0.0362 + 0.999i$
Analytic cond. $1.67647$
Root an. cond. $1.67647$
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  + (0.0275 + 0.999i)2-s + (0.635 + 0.771i)3-s + (−0.998 + 0.0550i)4-s + (0.993 − 0.110i)5-s + (−0.754 + 0.656i)6-s + (0.546 − 0.837i)7-s + (−0.0825 − 0.996i)8-s + (−0.191 + 0.981i)9-s + (0.137 + 0.990i)10-s + (0.945 + 0.324i)11-s + (−0.677 − 0.735i)12-s + (0.350 − 0.936i)13-s + (0.851 + 0.523i)14-s + (0.716 + 0.697i)15-s + (0.993 − 0.110i)16-s + (−0.998 − 0.0550i)17-s + ⋯
L(s)  = 1  + (0.0275 + 0.999i)2-s + (0.635 + 0.771i)3-s + (−0.998 + 0.0550i)4-s + (0.993 − 0.110i)5-s + (−0.754 + 0.656i)6-s + (0.546 − 0.837i)7-s + (−0.0825 − 0.996i)8-s + (−0.191 + 0.981i)9-s + (0.137 + 0.990i)10-s + (0.945 + 0.324i)11-s + (−0.677 − 0.735i)12-s + (0.350 − 0.936i)13-s + (0.851 + 0.523i)14-s + (0.716 + 0.697i)15-s + (0.993 − 0.110i)16-s + (−0.998 − 0.0550i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0362 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0362 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $-0.0362 + 0.999i$
Analytic conductor: \(1.67647\)
Root analytic conductor: \(1.67647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (0:\ ),\ -0.0362 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.302119461 + 1.350211900i\)
\(L(\frac12)\) \(\approx\) \(1.302119461 + 1.350211900i\)
\(L(1)\) \(\approx\) \(1.211698260 + 0.8514077491i\)
\(L(1)\) \(\approx\) \(1.211698260 + 0.8514077491i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (0.0275 + 0.999i)T \)
3 \( 1 + (0.635 + 0.771i)T \)
5 \( 1 + (0.993 - 0.110i)T \)
7 \( 1 + (0.546 - 0.837i)T \)
11 \( 1 + (0.945 + 0.324i)T \)
13 \( 1 + (0.350 - 0.936i)T \)
17 \( 1 + (-0.998 - 0.0550i)T \)
23 \( 1 + (0.635 - 0.771i)T \)
29 \( 1 + (0.451 + 0.892i)T \)
31 \( 1 + (-0.879 + 0.475i)T \)
37 \( 1 + (0.945 + 0.324i)T \)
41 \( 1 + (-0.962 + 0.272i)T \)
43 \( 1 + (0.137 - 0.990i)T \)
47 \( 1 + (-0.754 + 0.656i)T \)
53 \( 1 + (-0.754 + 0.656i)T \)
59 \( 1 + (-0.962 + 0.272i)T \)
61 \( 1 + (-0.821 - 0.569i)T \)
67 \( 1 + (0.904 + 0.426i)T \)
71 \( 1 + (-0.821 + 0.569i)T \)
73 \( 1 + (-0.998 - 0.0550i)T \)
79 \( 1 + (0.137 - 0.990i)T \)
83 \( 1 + (-0.401 - 0.915i)T \)
89 \( 1 + (-0.998 + 0.0550i)T \)
97 \( 1 + (0.904 - 0.426i)T \)
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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

−24.563114355302157631971440637273, −23.709184250388995999410781303864, −22.465156652420379482762299513951, −21.55945581438278227288593060829, −21.12074893778900206935504156550, −20.03822645022357340126264002854, −19.20886175968903368437279129661, −18.4371284103864135118509572571, −17.80011426947171210854883476281, −16.95440405099253425186026020085, −15.078169196028414692703766756927, −14.28933364414661430434799768702, −13.597686693177917298594742301310, −12.85250084895224431666909642320, −11.71703079923027270925437391405, −11.16218597730029693737138288877, −9.47021814457468108058763989912, −9.13783706456273439731169874644, −8.23204011433950532984343675158, −6.65377304871714917085446955314, −5.75085106433531745347718678117, −4.36630117349407532587138621673, −3.06252887879769002703606654272, −2.01722335343977275253623412590, −1.460122410959600773431599246534, 1.44578216412024896132972868184, 3.170308633097954678032979831597, 4.40181772489157474089287167759, 5.03864296721379377776945323485, 6.303901549985735338909298795065, 7.30804336101341272175550652722, 8.51040540467043959317420930770, 9.11530017422272885953285376544, 10.13262945343934181247280489011, 10.88293570137983364400162222512, 12.82477672650853709881045555396, 13.593773526072670314279847073551, 14.387920201764736052172290926226, 14.928736290431452329294900042561, 16.05673875485913196643128677008, 16.96069610961923432701665205383, 17.51046439308868103047647549035, 18.4706370961625789496471135819, 19.97859062822345898751942500169, 20.47815979392208239939048896151, 21.6969948112345521082284349566, 22.22894828671561663214897087791, 23.19616513976647075017049156409, 24.40420206409945290977198569049, 25.10685286797963193744334053987

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