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Properties

Label 1-311-311.169-r0-0-0
Degree $1$
Conductor $311$
Sign $-0.889 + 0.457i$
Analytic cond. $1.44427$
Root an. cond. $1.44427$
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  + (−0.612 + 0.790i)2-s + (−0.612 − 0.790i)3-s + (−0.250 − 0.968i)4-s + (0.528 + 0.848i)5-s + 6-s + (−0.0506 + 0.998i)7-s + (0.918 + 0.394i)8-s + (−0.250 + 0.968i)9-s + (−0.994 − 0.101i)10-s + (−0.440 − 0.897i)11-s + (−0.612 + 0.790i)12-s + (−0.250 + 0.968i)13-s + (−0.758 − 0.651i)14-s + (0.347 − 0.937i)15-s + (−0.874 + 0.485i)16-s + (−0.994 − 0.101i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.790i)2-s + (−0.612 − 0.790i)3-s + (−0.250 − 0.968i)4-s + (0.528 + 0.848i)5-s + 6-s + (−0.0506 + 0.998i)7-s + (0.918 + 0.394i)8-s + (−0.250 + 0.968i)9-s + (−0.994 − 0.101i)10-s + (−0.440 − 0.897i)11-s + (−0.612 + 0.790i)12-s + (−0.250 + 0.968i)13-s + (−0.758 − 0.651i)14-s + (0.347 − 0.937i)15-s + (−0.874 + 0.485i)16-s + (−0.994 − 0.101i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.889 + 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.889 + 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $-0.889 + 0.457i$
Analytic conductor: \(1.44427\)
Root analytic conductor: \(1.44427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (0:\ ),\ -0.889 + 0.457i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1044782901 + 0.4317531898i\)
\(L(\frac12)\) \(\approx\) \(0.1044782901 + 0.4317531898i\)
\(L(1)\) \(\approx\) \(0.5097788378 + 0.2551073055i\)
\(L(1)\) \(\approx\) \(0.5097788378 + 0.2551073055i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (-0.612 + 0.790i)T \)
3 \( 1 + (-0.612 - 0.790i)T \)
5 \( 1 + (0.528 + 0.848i)T \)
7 \( 1 + (-0.0506 + 0.998i)T \)
11 \( 1 + (-0.440 - 0.897i)T \)
13 \( 1 + (-0.250 + 0.968i)T \)
17 \( 1 + (-0.994 - 0.101i)T \)
19 \( 1 + (0.528 - 0.848i)T \)
23 \( 1 + (0.918 + 0.394i)T \)
29 \( 1 + (-0.758 + 0.651i)T \)
31 \( 1 + (-0.994 + 0.101i)T \)
37 \( 1 + (-0.0506 - 0.998i)T \)
41 \( 1 + (-0.954 + 0.299i)T \)
43 \( 1 + (-0.0506 + 0.998i)T \)
47 \( 1 + (-0.758 + 0.651i)T \)
53 \( 1 + (-0.0506 + 0.998i)T \)
59 \( 1 + (-0.0506 + 0.998i)T \)
61 \( 1 + (0.528 - 0.848i)T \)
67 \( 1 + (-0.954 - 0.299i)T \)
71 \( 1 + (0.151 + 0.988i)T \)
73 \( 1 + (0.688 + 0.724i)T \)
79 \( 1 + (-0.954 + 0.299i)T \)
83 \( 1 + (0.347 + 0.937i)T \)
89 \( 1 + (-0.0506 - 0.998i)T \)
97 \( 1 + (0.151 + 0.988i)T \)
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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

−25.12378847231742902855003860390, −23.818637906258081539652858624390, −22.70408147603562831426999365778, −22.18680943498664138371105996210, −20.8291276052620027810135249202, −20.57455041641021306799015776470, −19.901395149928477735100214871272, −18.30020566078391067040035791288, −17.46616514639508725724541087020, −16.981201633014592378184309958058, −16.15863448731751602827137979119, −14.99419127717190611072479679419, −13.4002527706685967107817385445, −12.77979371147055829744655995167, −11.75727841229786255773275029168, −10.58847437405337467015849656378, −10.081484913843157156824872039352, −9.278242182861024181186018152414, −8.12565118696151685929473812311, −6.9135086256482606869727870084, −5.286271768287994487566619383450, −4.48987120315284388398274964666, −3.42655685232090322440391290, −1.80013117340619483018509092616, −0.37429581157510235489641186195, 1.65355477695450926081097049256, 2.71368783679427789490750304871, 5.055433195536207725309131760786, 5.81273218032557283525736964174, 6.6956532931226881601148951302, 7.36677058275260642278476177847, 8.71942808322587725644629145815, 9.48369289603136253811556898874, 11.05005445449578851820154654630, 11.27375353306759080216381526130, 12.98661657396843897579376989726, 13.78785594160485966646210407036, 14.73214992967901089530436990174, 15.78338023610530868923668576069, 16.66931812739661399816928477562, 17.70064487747843928397323802533, 18.32576353876071701525403300144, 18.88509761403416149850045718849, 19.68363497727190427130262627914, 21.59592088063963909802886816306, 22.13304774496178598365957322823, 23.13390258081456669049037778647, 24.160286075207427355214622618395, 24.61189928823559491862299801546, 25.5976455766410764661889233250

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