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Properties

Label 1-349-349.126-r0-0-0
Degree $1$
Conductor $349$
Sign $-0.215 + 0.976i$
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  + (−0.561 + 0.827i)2-s + (0.976 − 0.214i)3-s + (−0.370 − 0.928i)4-s + (−0.161 + 0.986i)5-s + (−0.370 + 0.928i)6-s + (0.796 + 0.605i)7-s + (0.976 + 0.214i)8-s + (0.907 − 0.419i)9-s + (−0.725 − 0.687i)10-s + (0.0541 + 0.998i)11-s + (−0.561 − 0.827i)12-s + (−0.947 + 0.319i)13-s + (−0.947 + 0.319i)14-s + (0.0541 + 0.998i)15-s + (−0.725 + 0.687i)16-s + (0.976 − 0.214i)17-s + ⋯
L(s)  = 1  + (−0.561 + 0.827i)2-s + (0.976 − 0.214i)3-s + (−0.370 − 0.928i)4-s + (−0.161 + 0.986i)5-s + (−0.370 + 0.928i)6-s + (0.796 + 0.605i)7-s + (0.976 + 0.214i)8-s + (0.907 − 0.419i)9-s + (−0.725 − 0.687i)10-s + (0.0541 + 0.998i)11-s + (−0.561 − 0.827i)12-s + (−0.947 + 0.319i)13-s + (−0.947 + 0.319i)14-s + (0.0541 + 0.998i)15-s + (−0.725 + 0.687i)16-s + (0.976 − 0.214i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(349\)
Sign: $-0.215 + 0.976i$
Analytic conductor: \(1.62074\)
Root analytic conductor: \(1.62074\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 349,\ (0:\ ),\ -0.215 + 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8496693784 + 1.058021380i\)
\(L(\frac12)\) \(\approx\) \(0.8496693784 + 1.058021380i\)
\(L(1)\) \(\approx\) \(0.9613650849 + 0.5927112817i\)
\(L(1)\) \(\approx\) \(0.9613650849 + 0.5927112817i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 \)
good2 \( 1 + (-0.561 + 0.827i)T \)
3 \( 1 + (0.976 - 0.214i)T \)
5 \( 1 + (-0.161 + 0.986i)T \)
7 \( 1 + (0.796 + 0.605i)T \)
11 \( 1 + (0.0541 + 0.998i)T \)
13 \( 1 + (-0.947 + 0.319i)T \)
17 \( 1 + (0.976 - 0.214i)T \)
19 \( 1 + (0.468 - 0.883i)T \)
23 \( 1 + (-0.947 + 0.319i)T \)
29 \( 1 + (-0.561 + 0.827i)T \)
31 \( 1 + (0.267 - 0.963i)T \)
37 \( 1 + (-0.947 + 0.319i)T \)
41 \( 1 + (0.976 + 0.214i)T \)
43 \( 1 + (0.0541 + 0.998i)T \)
47 \( 1 + (0.647 - 0.762i)T \)
53 \( 1 + (-0.994 + 0.108i)T \)
59 \( 1 + (0.976 - 0.214i)T \)
61 \( 1 + (-0.561 - 0.827i)T \)
67 \( 1 + (0.468 + 0.883i)T \)
71 \( 1 + (0.976 + 0.214i)T \)
73 \( 1 + (0.796 + 0.605i)T \)
79 \( 1 + (-0.994 - 0.108i)T \)
83 \( 1 + (-0.161 + 0.986i)T \)
89 \( 1 + (-0.856 + 0.515i)T \)
97 \( 1 + (-0.161 - 0.986i)T \)
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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

−24.622603276921516386843140502620, −24.03686389271471233944707374269, −22.57668170992069016487352052106, −21.36020785374347724657823970304, −20.94705221218572774327004054162, −20.18466694618915758924744276437, −19.477474320438636271882041794309, −18.69467451885827817642855243131, −17.45379037199882575562472311238, −16.67062540405246826439244990137, −15.86386386307745110109882127545, −14.30275208319594996693398765358, −13.80867689937086372023732680106, −12.61955208491740190825179500116, −11.92069951930394967950939118640, −10.60715411849680295157680633101, −9.85445340986108738610699273713, −8.859588737620530855436780187208, −7.99100893569741545778002244616, −7.61213582552799273672461358899, −5.31603022933905112868331752301, −4.19654735523371857523970078695, −3.43612732106217671871864055135, −2.03653459749627425741643253389, −0.99201825670023429383647806471, 1.71044692067070527487156595176, 2.63384810736051399824739451317, 4.22613851896581479212846058248, 5.38275168860253379180274528481, 6.82437153318073548493023028204, 7.47746380544767138273026740362, 8.143250674545528154054940892924, 9.46349045647654722831509029189, 9.90259312869629450342912846997, 11.29103055878743134550357886747, 12.43169399362234747221151792420, 13.93839167285014450486041045252, 14.4901950818888473112096008433, 15.10208924685475449125167044592, 15.81154127674391607472523712909, 17.32393707239489679475568521277, 18.09621388854271623506631445358, 18.712993208274595072850940550517, 19.555529424085963351491049212768, 20.3938826164283676133977940886, 21.65852336720423480304787885497, 22.53645998850393166221851053985, 23.70948710750468321560307229304, 24.381049961355851510716256028517, 25.22898893631682520687790746556

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