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Properties

Label 1-349-349.108-r0-0-0
Degree $1$
Conductor $349$
Sign $-0.752 + 0.658i$
Analytic cond. $1.62074$
Root an. cond. $1.62074$
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  + (0.126 + 0.992i)2-s + (0.989 − 0.143i)3-s + (−0.968 + 0.250i)4-s + (0.403 + 0.915i)5-s + (0.267 + 0.963i)6-s + (−0.817 + 0.576i)7-s + (−0.370 − 0.928i)8-s + (0.958 − 0.284i)9-s + (−0.856 + 0.515i)10-s + (0.468 − 0.883i)11-s + (−0.922 + 0.386i)12-s + (−0.302 + 0.953i)13-s + (−0.674 − 0.738i)14-s + (0.530 + 0.847i)15-s + (0.874 − 0.484i)16-s + (−0.370 + 0.928i)17-s + ⋯
L(s)  = 1  + (0.126 + 0.992i)2-s + (0.989 − 0.143i)3-s + (−0.968 + 0.250i)4-s + (0.403 + 0.915i)5-s + (0.267 + 0.963i)6-s + (−0.817 + 0.576i)7-s + (−0.370 − 0.928i)8-s + (0.958 − 0.284i)9-s + (−0.856 + 0.515i)10-s + (0.468 − 0.883i)11-s + (−0.922 + 0.386i)12-s + (−0.302 + 0.953i)13-s + (−0.674 − 0.738i)14-s + (0.530 + 0.847i)15-s + (0.874 − 0.484i)16-s + (−0.370 + 0.928i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.752 + 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.752 + 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(349\)
Sign: $-0.752 + 0.658i$
Analytic conductor: \(1.62074\)
Root analytic conductor: \(1.62074\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (108, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 349,\ (0:\ ),\ -0.752 + 0.658i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5684267567 + 1.513904947i\)
\(L(\frac12)\) \(\approx\) \(0.5684267567 + 1.513904947i\)
\(L(1)\) \(\approx\) \(0.9961338293 + 0.9173635549i\)
\(L(1)\) \(\approx\) \(0.9961338293 + 0.9173635549i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 \)
good2 \( 1 + (0.126 + 0.992i)T \)
3 \( 1 + (0.989 - 0.143i)T \)
5 \( 1 + (0.403 + 0.915i)T \)
7 \( 1 + (-0.817 + 0.576i)T \)
11 \( 1 + (0.468 - 0.883i)T \)
13 \( 1 + (-0.302 + 0.953i)T \)
17 \( 1 + (-0.370 + 0.928i)T \)
19 \( 1 + (0.197 + 0.980i)T \)
23 \( 1 + (-0.302 + 0.953i)T \)
29 \( 1 + (-0.922 - 0.386i)T \)
31 \( 1 + (0.647 - 0.762i)T \)
37 \( 1 + (0.976 - 0.214i)T \)
41 \( 1 + (-0.370 - 0.928i)T \)
43 \( 1 + (-0.999 + 0.0361i)T \)
47 \( 1 + (0.0541 + 0.998i)T \)
53 \( 1 + (-0.561 - 0.827i)T \)
59 \( 1 + (0.989 - 0.143i)T \)
61 \( 1 + (0.796 + 0.605i)T \)
67 \( 1 + (-0.947 + 0.319i)T \)
71 \( 1 + (0.989 + 0.143i)T \)
73 \( 1 + (-0.0901 - 0.995i)T \)
79 \( 1 + (-0.561 + 0.827i)T \)
83 \( 1 + (0.590 - 0.806i)T \)
89 \( 1 + (0.935 - 0.353i)T \)
97 \( 1 + (0.403 - 0.915i)T \)
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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

−24.662799391767396103872609369645, −23.495791155998596288631242117224, −22.44449381224413432089928792884, −21.77402479462223645068997304211, −20.54335572011251433813673157547, −20.103162199753275626867156324381, −19.7935409827492502965375400178, −18.46513817495479619818962780516, −17.56434646252133502434913555340, −16.491772547663771033729631264283, −15.342113856870767035527157304289, −14.31223545012065623286531615844, −13.283739080479983199892566494551, −12.981924694357010748038288414287, −11.96916735977230114904399611671, −10.438952553318100931130961026759, −9.690586084934658008614265564984, −9.161386683214933237051000974208, −8.09076007669693735506345240687, −6.78821250297508233971539205253, −5.04380200191373626944727563139, −4.34523454639816128728850623159, −3.16759617446646944936233458797, −2.22463589353043369995869473544, −0.88946707982117619087669671285, 1.96438985563310966199533545409, 3.31678062484930410321123256517, 3.974485254741277515653833956116, 5.852575216933681363380549851697, 6.42651332434261784180237931891, 7.41389744147500644340871579050, 8.44883848862076719166964372688, 9.397234995162583958139327331664, 9.97627554552989548091565083778, 11.6841049246631771596167028417, 12.980619361337757513103971652190, 13.683823167430404841395275813935, 14.48074165959023766868753871077, 15.12606117525543543426266835456, 16.05832787635714116974765601840, 17.02093405950809456722336688916, 18.20536150080498371304707103151, 19.10177302080296458154521947671, 19.264775661152347220055889518645, 21.08971321022178393865356603362, 21.8928417765067034144309459448, 22.3818763603939827322997626360, 23.69146807849548842406069130874, 24.47544967247553410904770385599, 25.329660061106586258967550471909

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