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Properties

Label 1-349-349.111-r0-0-0
Degree $1$
Conductor $349$
Sign $-0.998 - 0.0509i$
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  + (−0.922 + 0.386i)2-s + (−0.619 + 0.785i)3-s + (0.700 − 0.713i)4-s + (0.590 + 0.806i)5-s + (0.267 − 0.963i)6-s + (−0.0901 + 0.995i)7-s + (−0.370 + 0.928i)8-s + (−0.232 − 0.972i)9-s + (−0.856 − 0.515i)10-s + (0.468 + 0.883i)11-s + (0.126 + 0.992i)12-s + (−0.674 + 0.738i)13-s + (−0.302 − 0.953i)14-s + (−0.999 − 0.0361i)15-s + (−0.0180 − 0.999i)16-s + (−0.370 − 0.928i)17-s + ⋯
L(s)  = 1  + (−0.922 + 0.386i)2-s + (−0.619 + 0.785i)3-s + (0.700 − 0.713i)4-s + (0.590 + 0.806i)5-s + (0.267 − 0.963i)6-s + (−0.0901 + 0.995i)7-s + (−0.370 + 0.928i)8-s + (−0.232 − 0.972i)9-s + (−0.856 − 0.515i)10-s + (0.468 + 0.883i)11-s + (0.126 + 0.992i)12-s + (−0.674 + 0.738i)13-s + (−0.302 − 0.953i)14-s + (−0.999 − 0.0361i)15-s + (−0.0180 − 0.999i)16-s + (−0.370 − 0.928i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(349\)
Sign: $-0.998 - 0.0509i$
Analytic conductor: \(1.62074\)
Root analytic conductor: \(1.62074\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 349,\ (0:\ ),\ -0.998 - 0.0509i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01552801453 + 0.6095038666i\)
\(L(\frac12)\) \(\approx\) \(0.01552801453 + 0.6095038666i\)
\(L(1)\) \(\approx\) \(0.4312725408 + 0.4387422971i\)
\(L(1)\) \(\approx\) \(0.4312725408 + 0.4387422971i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 \)
good2 \( 1 + (-0.922 + 0.386i)T \)
3 \( 1 + (-0.619 + 0.785i)T \)
5 \( 1 + (0.590 + 0.806i)T \)
7 \( 1 + (-0.0901 + 0.995i)T \)
11 \( 1 + (0.468 + 0.883i)T \)
13 \( 1 + (-0.674 + 0.738i)T \)
17 \( 1 + (-0.370 - 0.928i)T \)
19 \( 1 + (0.750 + 0.661i)T \)
23 \( 1 + (-0.674 + 0.738i)T \)
29 \( 1 + (0.126 - 0.992i)T \)
31 \( 1 + (0.647 + 0.762i)T \)
37 \( 1 + (0.976 + 0.214i)T \)
41 \( 1 + (-0.370 + 0.928i)T \)
43 \( 1 + (0.530 - 0.847i)T \)
47 \( 1 + (0.0541 - 0.998i)T \)
53 \( 1 + (-0.561 + 0.827i)T \)
59 \( 1 + (-0.619 + 0.785i)T \)
61 \( 1 + (0.796 - 0.605i)T \)
67 \( 1 + (-0.947 - 0.319i)T \)
71 \( 1 + (-0.619 - 0.785i)T \)
73 \( 1 + (-0.817 - 0.576i)T \)
79 \( 1 + (-0.561 - 0.827i)T \)
83 \( 1 + (0.403 - 0.915i)T \)
89 \( 1 + (-0.773 + 0.633i)T \)
97 \( 1 + (0.590 - 0.806i)T \)
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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

−24.25607997729492177555473519696, −24.058350620800701642175140423172, −22.38847709384909570638471828869, −21.808422313827368184659702089149, −20.51169749293819732433236795389, −19.848693249305284787054265507274, −19.15851554128937782339397857668, −17.87102069835040370363077976709, −17.42735601123730915345590448003, −16.69111489791511373022016220297, −16.03220551715913236215464440851, −14.192644678163957895641518248727, −13.13386443198331209176480365464, −12.59975980685416552457901164096, −11.48620191422854184066149687081, −10.63306461258133251486522479015, −9.760106481980881625842992323630, −8.54301125224849660185549872181, −7.7599270255067789776977882479, −6.68242213572392341877835501705, −5.79095737055489382016418046316, −4.341233560579068425374561490077, −2.739545845437246436438939013082, −1.41181250232535714550533326313, −0.57735822543119790061137885663, 1.80831068562398485518336142573, 2.92190784748828757342783495597, 4.70723075504524830526398653117, 5.77457254065531747773299486108, 6.50732539019340581445337699977, 7.49144758256490031321218542030, 9.11671044275398780637750338477, 9.64082220051024562633628412328, 10.2667719565662574579440007437, 11.69742594383346578138777631669, 11.869830080477866459451296478934, 13.99098400189889647353243079399, 14.86531330056003713421927950924, 15.510522779226052412531010704487, 16.41320637211432677394280644082, 17.41686782882131010292558428986, 18.001779649102988852511409969698, 18.75319779187269097768782248606, 19.89027594862005178873959379990, 20.936993049702211864291133524986, 21.85129771814959092279764343080, 22.539848350832440888935147726935, 23.47033432198439877050912965506, 24.84116489185671819673861361033, 25.279867007274143444583125419276

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