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Properties

Label 1-349-349.14-r0-0-0
Degree $1$
Conductor $349$
Sign $-0.930 + 0.365i$
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  + (0.336 + 0.941i)2-s + (0.874 + 0.484i)3-s + (−0.773 + 0.633i)4-s + (−0.619 + 0.785i)5-s + (−0.161 + 0.986i)6-s + (0.837 + 0.546i)7-s + (−0.856 − 0.515i)8-s + (0.530 + 0.847i)9-s + (−0.947 − 0.319i)10-s + (0.796 + 0.605i)11-s + (−0.983 + 0.179i)12-s + (0.958 − 0.284i)13-s + (−0.232 + 0.972i)14-s + (−0.922 + 0.386i)15-s + (0.197 − 0.980i)16-s + (−0.856 + 0.515i)17-s + ⋯
L(s)  = 1  + (0.336 + 0.941i)2-s + (0.874 + 0.484i)3-s + (−0.773 + 0.633i)4-s + (−0.619 + 0.785i)5-s + (−0.161 + 0.986i)6-s + (0.837 + 0.546i)7-s + (−0.856 − 0.515i)8-s + (0.530 + 0.847i)9-s + (−0.947 − 0.319i)10-s + (0.796 + 0.605i)11-s + (−0.983 + 0.179i)12-s + (0.958 − 0.284i)13-s + (−0.232 + 0.972i)14-s + (−0.922 + 0.386i)15-s + (0.197 − 0.980i)16-s + (−0.856 + 0.515i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(349\)
Sign: $-0.930 + 0.365i$
Analytic conductor: \(1.62074\)
Root analytic conductor: \(1.62074\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 349,\ (0:\ ),\ -0.930 + 0.365i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3484299104 + 1.842228352i\)
\(L(\frac12)\) \(\approx\) \(0.3484299104 + 1.842228352i\)
\(L(1)\) \(\approx\) \(0.9342802086 + 1.193785505i\)
\(L(1)\) \(\approx\) \(0.9342802086 + 1.193785505i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 \)
good2 \( 1 + (0.336 + 0.941i)T \)
3 \( 1 + (0.874 + 0.484i)T \)
5 \( 1 + (-0.619 + 0.785i)T \)
7 \( 1 + (0.837 + 0.546i)T \)
11 \( 1 + (0.796 + 0.605i)T \)
13 \( 1 + (0.958 - 0.284i)T \)
17 \( 1 + (-0.856 + 0.515i)T \)
19 \( 1 + (-0.0901 - 0.995i)T \)
23 \( 1 + (0.958 - 0.284i)T \)
29 \( 1 + (-0.983 - 0.179i)T \)
31 \( 1 + (-0.994 + 0.108i)T \)
37 \( 1 + (-0.725 - 0.687i)T \)
41 \( 1 + (-0.856 - 0.515i)T \)
43 \( 1 + (0.126 - 0.992i)T \)
47 \( 1 + (-0.561 - 0.827i)T \)
53 \( 1 + (0.267 + 0.963i)T \)
59 \( 1 + (0.874 + 0.484i)T \)
61 \( 1 + (0.647 + 0.762i)T \)
67 \( 1 + (0.907 - 0.419i)T \)
71 \( 1 + (0.874 - 0.484i)T \)
73 \( 1 + (-0.891 + 0.452i)T \)
79 \( 1 + (0.267 - 0.963i)T \)
83 \( 1 + (0.989 + 0.143i)T \)
89 \( 1 + (-0.302 - 0.953i)T \)
97 \( 1 + (-0.619 - 0.785i)T \)
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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

−24.236885912795083399375073943956, −23.70058230090772303992488085512, −22.77789794299300619699933428453, −21.43388050462112942189685169722, −20.592123605738969696042857496421, −20.3014012274714105490279107736, −19.28873244556864954614110260636, −18.63621086724323491445734327475, −17.58320168927961391053899735683, −16.39706333034855064995031558975, −15.07843774740797140369459501764, −14.29403947361317968157503979646, −13.45248303842417113335085573051, −12.79572406231817122966153638253, −11.57405004209402065227699804753, −11.139892810194787963269242140442, −9.533275166381053726037984607061, −8.719492886413033955772503621338, −8.08378856634518482908362949823, −6.68967624040650675304659865206, −5.15986241211698231151793391581, −3.99086813461635893347881524309, −3.4884926625223924911826614476, −1.765305821281118446240280082993, −1.08575152408319322310044569902, 2.16018006092667035057036416666, 3.51383762838593581170555444420, 4.21305388254385324441958035551, 5.30584730528581395778564017421, 6.75761610790688536741880249139, 7.47765161605782350809644581114, 8.66174515280331587435551160599, 8.98767017931686771569075900886, 10.598104727408591858805170692204, 11.53564114661396365261732787892, 12.85922310219809001957942449524, 13.83364952794165801844509598175, 14.826715116589450507048836112238, 15.130256582580225376586947422399, 15.82496506028788049977203726226, 17.123474908199344737050975968788, 18.09823453278649221748638975919, 18.88761234879608307185508214124, 19.979352225404246029423136462172, 20.97250359217621041401430454638, 21.96108215586223909040531630810, 22.46900352572226614815809769723, 23.6043976194625562473159496950, 24.45935147933912690323338345985, 25.297541262439319843993377870872

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