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Properties

Label 1-812-812.55-r1-0-0
Degree $1$
Conductor $812$
Sign $-0.995 + 0.0915i$
Analytic cond. $87.2615$
Root an. cond. $87.2615$
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  + (−0.781 + 0.623i)3-s + (−0.900 + 0.433i)5-s + (0.222 − 0.974i)9-s + (−0.974 + 0.222i)11-s + (−0.222 − 0.974i)13-s + (0.433 − 0.900i)15-s − i·17-s + (0.781 + 0.623i)19-s + (0.900 + 0.433i)23-s + (0.623 − 0.781i)25-s + (0.433 + 0.900i)27-s + (−0.433 − 0.900i)31-s + (0.623 − 0.781i)33-s + (0.974 + 0.222i)37-s + (0.781 + 0.623i)39-s + ⋯
L(s)  = 1  + (−0.781 + 0.623i)3-s + (−0.900 + 0.433i)5-s + (0.222 − 0.974i)9-s + (−0.974 + 0.222i)11-s + (−0.222 − 0.974i)13-s + (0.433 − 0.900i)15-s − i·17-s + (0.781 + 0.623i)19-s + (0.900 + 0.433i)23-s + (0.623 − 0.781i)25-s + (0.433 + 0.900i)27-s + (−0.433 − 0.900i)31-s + (0.623 − 0.781i)33-s + (0.974 + 0.222i)37-s + (0.781 + 0.623i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 + 0.0915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 + 0.0915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(812\)    =    \(2^{2} \cdot 7 \cdot 29\)
Sign: $-0.995 + 0.0915i$
Analytic conductor: \(87.2615\)
Root analytic conductor: \(87.2615\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{812} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 812,\ (1:\ ),\ -0.995 + 0.0915i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01189793904 + 0.2593388985i\)
\(L(\frac12)\) \(\approx\) \(0.01189793904 + 0.2593388985i\)
\(L(1)\) \(\approx\) \(0.6030885426 + 0.1295193452i\)
\(L(1)\) \(\approx\) \(0.6030885426 + 0.1295193452i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
29 \( 1 \)
good3 \( 1 + (-0.781 + 0.623i)T \)
5 \( 1 + (-0.900 + 0.433i)T \)
11 \( 1 + (-0.974 + 0.222i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 - iT \)
19 \( 1 + (0.781 + 0.623i)T \)
23 \( 1 + (0.900 + 0.433i)T \)
31 \( 1 + (-0.433 - 0.900i)T \)
37 \( 1 + (0.974 + 0.222i)T \)
41 \( 1 + iT \)
43 \( 1 + (-0.433 + 0.900i)T \)
47 \( 1 + (-0.974 + 0.222i)T \)
53 \( 1 + (-0.900 + 0.433i)T \)
59 \( 1 + T \)
61 \( 1 + (0.781 - 0.623i)T \)
67 \( 1 + (-0.222 + 0.974i)T \)
71 \( 1 + (-0.222 - 0.974i)T \)
73 \( 1 + (0.433 - 0.900i)T \)
79 \( 1 + (0.974 + 0.222i)T \)
83 \( 1 + (0.623 - 0.781i)T \)
89 \( 1 + (0.433 + 0.900i)T \)
97 \( 1 + (0.781 + 0.623i)T \)
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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

−21.71621089365195604679499152760, −20.90110148115660027210178184313, −19.81730887235591497566009761018, −19.16214116434549495223776632890, −18.52780855344405570563737824894, −17.62201933986011042985582837842, −16.72448117065535928005557700193, −16.16288282479164051242936558187, −15.37216154218805239012284891937, −14.27049468897158860306826242470, −13.1498983443521795830913559804, −12.68772380885501107016915199676, −11.75062268274024621469867244886, −11.15187606369090883294235523978, −10.33158081146837545215552737714, −8.99175109559655239974952274103, −8.148467639677118404650507519945, −7.30973944395244817235670620514, −6.61204705808339150667778064284, −5.349559174737439058628723924732, −4.79406102411802986760500063431, −3.63604727293965786597545085588, −2.3157101287846803667257919856, −1.11261541151883082701787560129, −0.093903469645713564127230550629, 0.877648178308628961140018010830, 2.807227869143335964205996319211, 3.47629083181154333797197866062, 4.70447145390855243551588463479, 5.27303168299133114570770270934, 6.35915989929790730115746900850, 7.46624649420606383327471772937, 7.991790060057767886727127439426, 9.46219049330155213698681073179, 10.09812593769381545415578859287, 11.06671559767390903462635590743, 11.52657621869702904547195051808, 12.48774715752412486735223252707, 13.29086477692461430088922044552, 14.74023704365048221198103087590, 15.165803920260762023349653004291, 16.06177871981021402394213027697, 16.49427462959319244702230477410, 17.8017897607726494186460538985, 18.203386213390163237899503127941, 19.11178199757133025569816344333, 20.31979116630743055240360845098, 20.65985974181877157057556062176, 21.79334970674517111572957802465, 22.56982120430834923766344521660

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