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Properties

Label 1-349-349.148-r0-0-0
Degree $1$
Conductor $349$
Sign $-0.991 + 0.131i$
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  + (−0.674 + 0.738i)2-s + (−0.922 − 0.386i)3-s + (−0.0901 − 0.995i)4-s + (−0.999 + 0.0361i)5-s + (0.907 − 0.419i)6-s + (−0.619 − 0.785i)7-s + (0.796 + 0.605i)8-s + (0.700 + 0.713i)9-s + (0.647 − 0.762i)10-s + (−0.161 − 0.986i)11-s + (−0.302 + 0.953i)12-s + (−0.436 − 0.899i)13-s + (0.997 + 0.0721i)14-s + (0.935 + 0.353i)15-s + (−0.983 + 0.179i)16-s + (0.796 − 0.605i)17-s + ⋯
L(s)  = 1  + (−0.674 + 0.738i)2-s + (−0.922 − 0.386i)3-s + (−0.0901 − 0.995i)4-s + (−0.999 + 0.0361i)5-s + (0.907 − 0.419i)6-s + (−0.619 − 0.785i)7-s + (0.796 + 0.605i)8-s + (0.700 + 0.713i)9-s + (0.647 − 0.762i)10-s + (−0.161 − 0.986i)11-s + (−0.302 + 0.953i)12-s + (−0.436 − 0.899i)13-s + (0.997 + 0.0721i)14-s + (0.935 + 0.353i)15-s + (−0.983 + 0.179i)16-s + (0.796 − 0.605i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(349\)
Sign: $-0.991 + 0.131i$
Analytic conductor: \(1.62074\)
Root analytic conductor: \(1.62074\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 349,\ (0:\ ),\ -0.991 + 0.131i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.005149500052 - 0.07787639801i\)
\(L(\frac12)\) \(\approx\) \(0.005149500052 - 0.07787639801i\)
\(L(1)\) \(\approx\) \(0.3697778026 - 0.03229738661i\)
\(L(1)\) \(\approx\) \(0.3697778026 - 0.03229738661i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 \)
good2 \( 1 + (-0.674 + 0.738i)T \)
3 \( 1 + (-0.922 - 0.386i)T \)
5 \( 1 + (-0.999 + 0.0361i)T \)
7 \( 1 + (-0.619 - 0.785i)T \)
11 \( 1 + (-0.161 - 0.986i)T \)
13 \( 1 + (-0.436 - 0.899i)T \)
17 \( 1 + (0.796 - 0.605i)T \)
19 \( 1 + (0.590 + 0.806i)T \)
23 \( 1 + (-0.436 - 0.899i)T \)
29 \( 1 + (-0.302 - 0.953i)T \)
31 \( 1 + (-0.725 + 0.687i)T \)
37 \( 1 + (-0.561 + 0.827i)T \)
41 \( 1 + (0.796 + 0.605i)T \)
43 \( 1 + (-0.773 + 0.633i)T \)
47 \( 1 + (-0.856 - 0.515i)T \)
53 \( 1 + (-0.947 + 0.319i)T \)
59 \( 1 + (-0.922 - 0.386i)T \)
61 \( 1 + (0.976 - 0.214i)T \)
67 \( 1 + (-0.994 - 0.108i)T \)
71 \( 1 + (-0.922 + 0.386i)T \)
73 \( 1 + (0.989 - 0.143i)T \)
79 \( 1 + (-0.947 - 0.319i)T \)
83 \( 1 + (0.530 + 0.847i)T \)
89 \( 1 + (0.837 - 0.546i)T \)
97 \( 1 + (-0.999 - 0.0361i)T \)
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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

−25.72842463543994626956371148325, −24.24572458253670309943319349533, −23.355810613520259082705183820710, −22.40610125094423251421242263717, −21.859773862692639252334066206282, −20.896970661310942879500878261955, −19.842439224091075494781936673561, −19.09606557294624456488648924619, −18.273172813538620817818677583405, −17.39044255210003486532791796525, −16.365340464243969054507064355100, −15.827910346922440317727608934646, −14.822590876450587503106082006997, −12.92516645063744869209052788212, −12.21161272248873216284776340146, −11.709547764730147212631320250232, −10.75541663545385575400328436144, −9.6616534951724182999766812358, −9.08899906450219820290513432887, −7.598722840666215545335146321540, −6.86692904161839933507012476568, −5.30560381534297452317189843409, −4.19837672336139693185947214855, −3.27420583711147737076916234685, −1.728152144582505979468776375989, 0.08434974518360673246644192926, 1.06944056076686308711311576406, 3.2566503861714422951128981387, 4.69541993658348994659597527366, 5.73431697182140097941923749598, 6.67573555549826451561092888961, 7.64775476793691348937927412192, 8.121753183588388114709069843941, 9.806524918233667850073461946590, 10.536748785590313477163740452395, 11.420370720990942360762656608674, 12.477671788205094163832876514151, 13.58192858417835729429989154321, 14.63598360730671015786390325984, 15.95705112704656300347090601591, 16.3163891187797421857625867598, 17.02919249474494783012537494102, 18.208379620463928992072866737233, 18.84992710178465989092125380607, 19.61479614712151930828332153093, 20.5344077051824968072685344661, 22.295916241524563921676513415746, 22.9358402516753735110911463074, 23.48574867374341514150194573844, 24.39346255198222353657706846115

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