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Properties

Label 1-367-367.137-r0-0-0
Degree $1$
Conductor $367$
Sign $-0.528 + 0.848i$
Analytic cond. $1.70434$
Root an. cond. $1.70434$
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  + (0.679 + 0.733i)2-s + (−0.279 − 0.960i)3-s + (−0.0771 + 0.997i)4-s + (0.128 + 0.991i)5-s + (0.514 − 0.857i)6-s + (0.978 + 0.204i)7-s + (−0.784 + 0.620i)8-s + (−0.843 + 0.536i)9-s + (−0.640 + 0.767i)10-s + (−0.179 + 0.983i)11-s + (0.978 − 0.204i)12-s + (−0.843 + 0.536i)13-s + (0.514 + 0.857i)14-s + (0.916 − 0.400i)15-s + (−0.988 − 0.153i)16-s + (−0.716 − 0.697i)17-s + ⋯
L(s)  = 1  + (0.679 + 0.733i)2-s + (−0.279 − 0.960i)3-s + (−0.0771 + 0.997i)4-s + (0.128 + 0.991i)5-s + (0.514 − 0.857i)6-s + (0.978 + 0.204i)7-s + (−0.784 + 0.620i)8-s + (−0.843 + 0.536i)9-s + (−0.640 + 0.767i)10-s + (−0.179 + 0.983i)11-s + (0.978 − 0.204i)12-s + (−0.843 + 0.536i)13-s + (0.514 + 0.857i)14-s + (0.916 − 0.400i)15-s + (−0.988 − 0.153i)16-s + (−0.716 − 0.697i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.528 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.528 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(367\)
Sign: $-0.528 + 0.848i$
Analytic conductor: \(1.70434\)
Root analytic conductor: \(1.70434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{367} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 367,\ (0:\ ),\ -0.528 + 0.848i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7237720051 + 1.303645111i\)
\(L(\frac12)\) \(\approx\) \(0.7237720051 + 1.303645111i\)
\(L(1)\) \(\approx\) \(1.107135740 + 0.6812870989i\)
\(L(1)\) \(\approx\) \(1.107135740 + 0.6812870989i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad367 \( 1 \)
good2 \( 1 + (0.679 + 0.733i)T \)
3 \( 1 + (-0.279 - 0.960i)T \)
5 \( 1 + (0.128 + 0.991i)T \)
7 \( 1 + (0.978 + 0.204i)T \)
11 \( 1 + (-0.179 + 0.983i)T \)
13 \( 1 + (-0.843 + 0.536i)T \)
17 \( 1 + (-0.716 - 0.697i)T \)
19 \( 1 + (0.870 - 0.492i)T \)
23 \( 1 + (-0.988 + 0.153i)T \)
29 \( 1 + (0.952 - 0.304i)T \)
31 \( 1 + (-0.0771 + 0.997i)T \)
37 \( 1 + (-0.558 + 0.829i)T \)
41 \( 1 + (-0.179 - 0.983i)T \)
43 \( 1 + (0.870 + 0.492i)T \)
47 \( 1 + (0.994 + 0.102i)T \)
53 \( 1 + (0.128 + 0.991i)T \)
59 \( 1 + (-0.894 - 0.447i)T \)
61 \( 1 + (0.514 + 0.857i)T \)
67 \( 1 + (-0.894 + 0.447i)T \)
71 \( 1 + (-0.0771 - 0.997i)T \)
73 \( 1 + (0.815 - 0.579i)T \)
79 \( 1 + (0.916 + 0.400i)T \)
83 \( 1 + T \)
89 \( 1 + (0.994 + 0.102i)T \)
97 \( 1 + (-0.988 + 0.153i)T \)
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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

−24.20383725525336139506500798912, −23.49131396315254468343277054097, −22.267548907928639100796317497543, −21.70081097002333847391543073074, −20.93320811970791317672857266156, −20.27252069102653995486482048902, −19.59661698160767888782482594578, −18.08312698199796757284958297796, −17.26102131005147842061285517655, −16.22117683157588869208375928463, −15.39873158732581452010979261388, −14.378631405067794306954682683284, −13.672136106653356578720291379999, −12.43427693575627987496762258162, −11.69764078902101274910407040333, −10.790324351665924201319071446329, −10.01032125366600072163397356992, −8.97565850669385376525062916869, −8.031761982484806456695054564651, −5.9904172095398611320596313989, −5.29942191405860108806590917245, −4.51276490526894019601552931296, −3.64894801637939866111930308857, −2.21235360401013710047163534448, −0.73364358681409595373864476105, 2.03404178750600784939084843714, 2.77289707291411899756305793045, 4.512380528059808996230434380796, 5.3397661706763280151491569581, 6.53099890996205300271544334636, 7.25603219965203400275319519515, 7.80378257732279188499293967434, 9.14961378178545259233546229633, 10.705102829620800488345722024325, 11.907767252220275486870303266906, 12.12172029770188932637995790822, 13.74539758158869518863335734445, 14.01296120394549996660339192084, 14.993480808709571451544309649247, 15.86569180752743003215981399382, 17.329822778003091258194859302038, 17.78317689212995594433131733026, 18.35276566824948899106307827455, 19.66033205103386633491404097565, 20.71454324970625553013586114882, 21.97051480527685943452610923758, 22.37897878907171448857785870934, 23.3702879540167931921160670398, 24.06571190192237681030222437906, 24.802832376241144837666628897

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