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Properties

Label 1-349-349.12-r0-0-0
Degree $1$
Conductor $349$
Sign $-0.592 + 0.805i$
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  + (0.874 + 0.484i)2-s + (0.837 + 0.546i)3-s + (0.530 + 0.847i)4-s + (−0.0901 + 0.995i)5-s + (0.468 + 0.883i)6-s + (−0.773 + 0.633i)7-s + (0.0541 + 0.998i)8-s + (0.403 + 0.915i)9-s + (−0.561 + 0.827i)10-s + (−0.370 − 0.928i)11-s + (−0.0180 + 0.999i)12-s + (0.336 − 0.941i)13-s + (−0.983 + 0.179i)14-s + (−0.619 + 0.785i)15-s + (−0.436 + 0.899i)16-s + (0.0541 − 0.998i)17-s + ⋯
L(s)  = 1  + (0.874 + 0.484i)2-s + (0.837 + 0.546i)3-s + (0.530 + 0.847i)4-s + (−0.0901 + 0.995i)5-s + (0.468 + 0.883i)6-s + (−0.773 + 0.633i)7-s + (0.0541 + 0.998i)8-s + (0.403 + 0.915i)9-s + (−0.561 + 0.827i)10-s + (−0.370 − 0.928i)11-s + (−0.0180 + 0.999i)12-s + (0.336 − 0.941i)13-s + (−0.983 + 0.179i)14-s + (−0.619 + 0.785i)15-s + (−0.436 + 0.899i)16-s + (0.0541 − 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(349\)
Sign: $-0.592 + 0.805i$
Analytic conductor: \(1.62074\)
Root analytic conductor: \(1.62074\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 349,\ (0:\ ),\ -0.592 + 0.805i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.145335156 + 2.262605467i\)
\(L(\frac12)\) \(\approx\) \(1.145335156 + 2.262605467i\)
\(L(1)\) \(\approx\) \(1.509519806 + 1.301423841i\)
\(L(1)\) \(\approx\) \(1.509519806 + 1.301423841i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 \)
good2 \( 1 + (0.874 + 0.484i)T \)
3 \( 1 + (0.837 + 0.546i)T \)
5 \( 1 + (-0.0901 + 0.995i)T \)
7 \( 1 + (-0.773 + 0.633i)T \)
11 \( 1 + (-0.370 - 0.928i)T \)
13 \( 1 + (0.336 - 0.941i)T \)
17 \( 1 + (0.0541 - 0.998i)T \)
19 \( 1 + (0.700 + 0.713i)T \)
23 \( 1 + (0.336 - 0.941i)T \)
29 \( 1 + (-0.0180 - 0.999i)T \)
31 \( 1 + (-0.947 - 0.319i)T \)
37 \( 1 + (0.647 + 0.762i)T \)
41 \( 1 + (0.0541 + 0.998i)T \)
43 \( 1 + (0.989 + 0.143i)T \)
47 \( 1 + (0.976 + 0.214i)T \)
53 \( 1 + (-0.725 + 0.687i)T \)
59 \( 1 + (0.837 + 0.546i)T \)
61 \( 1 + (-0.856 - 0.515i)T \)
67 \( 1 + (0.267 + 0.963i)T \)
71 \( 1 + (0.837 - 0.546i)T \)
73 \( 1 + (0.935 + 0.353i)T \)
79 \( 1 + (-0.725 - 0.687i)T \)
83 \( 1 + (-0.817 - 0.576i)T \)
89 \( 1 + (0.126 + 0.992i)T \)
97 \( 1 + (-0.0901 - 0.995i)T \)
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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

−24.05541843665326619540960771854, −23.87224060481168620427786665631, −22.99273723931722877504336332239, −21.6701707721140241649731709155, −20.89561120530918406701776765874, −19.974793199795964457267211497899, −19.73686238203370944410230301799, −18.70729706785159120652475925355, −17.422787963604844753142324582, −16.144878095162700466668381162525, −15.45693263079346984333625822925, −14.30217150189788835886772372015, −13.498392971767577318392487492969, −12.78669134709506909861914853621, −12.29242777656238652472213139354, −10.94519454199655469853024482319, −9.63312106328322017580818641849, −9.08068544796272913546750442578, −7.51097945814958745820744352463, −6.797062780380783883398083546209, −5.47295356177546877920322641304, −4.204083338502174067500919818046, −3.535421122472449901788638933797, −2.12325667330330870961305506730, −1.163062135434217391962197898336, 2.71128061116513014966226574592, 2.93738917958604315652453766522, 3.96600268913100538521415640130, 5.42521735743121585060514558923, 6.21964494619659606002116905701, 7.489164620461701890733682514791, 8.22480656115675169071868399936, 9.45450392704881183817720937850, 10.56905450162564164298267522805, 11.49751394460314487653196585496, 12.79685103177638269750160395232, 13.63987203931032403132656230914, 14.38138217593399020995427386919, 15.28486259532276358606419280061, 15.851334121280986697652064372182, 16.6014136560386505547507884933, 18.28251392778936600064345214051, 18.89675771794445273987093628299, 20.11654392307845680696885722573, 20.89012851582135269531368051378, 21.87658493291759161652428651230, 22.43598796924393800234513840513, 23.08744367224582831303906343188, 24.527309292355805521349800081666, 25.17571054172085749432861662445

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