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Properties

Label 1-349-349.115-r0-0-0
Degree $1$
Conductor $349$
Sign $-0.431 + 0.902i$
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  + (−0.907 + 0.419i)2-s + (0.267 + 0.963i)3-s + (0.647 − 0.762i)4-s + (−0.561 + 0.827i)5-s + (−0.647 − 0.762i)6-s + (0.725 − 0.687i)7-s + (−0.267 + 0.963i)8-s + (−0.856 + 0.515i)9-s + (0.161 − 0.986i)10-s + (0.947 + 0.319i)11-s + (0.907 + 0.419i)12-s + (0.370 − 0.928i)13-s + (−0.370 + 0.928i)14-s + (−0.947 − 0.319i)15-s + (−0.161 − 0.986i)16-s + (0.267 + 0.963i)17-s + ⋯
L(s)  = 1  + (−0.907 + 0.419i)2-s + (0.267 + 0.963i)3-s + (0.647 − 0.762i)4-s + (−0.561 + 0.827i)5-s + (−0.647 − 0.762i)6-s + (0.725 − 0.687i)7-s + (−0.267 + 0.963i)8-s + (−0.856 + 0.515i)9-s + (0.161 − 0.986i)10-s + (0.947 + 0.319i)11-s + (0.907 + 0.419i)12-s + (0.370 − 0.928i)13-s + (−0.370 + 0.928i)14-s + (−0.947 − 0.319i)15-s + (−0.161 − 0.986i)16-s + (0.267 + 0.963i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.431 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.431 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(349\)
Sign: $-0.431 + 0.902i$
Analytic conductor: \(1.62074\)
Root analytic conductor: \(1.62074\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 349,\ (0:\ ),\ -0.431 + 0.902i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4982570436 + 0.7905158814i\)
\(L(\frac12)\) \(\approx\) \(0.4982570436 + 0.7905158814i\)
\(L(1)\) \(\approx\) \(0.6738723203 + 0.4665172287i\)
\(L(1)\) \(\approx\) \(0.6738723203 + 0.4665172287i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 \)
good2 \( 1 + (-0.907 + 0.419i)T \)
3 \( 1 + (0.267 + 0.963i)T \)
5 \( 1 + (-0.561 + 0.827i)T \)
7 \( 1 + (0.725 - 0.687i)T \)
11 \( 1 + (0.947 + 0.319i)T \)
13 \( 1 + (0.370 - 0.928i)T \)
17 \( 1 + (0.267 + 0.963i)T \)
19 \( 1 + (0.976 + 0.214i)T \)
23 \( 1 + (-0.370 + 0.928i)T \)
29 \( 1 + (0.907 - 0.419i)T \)
31 \( 1 + (0.0541 + 0.998i)T \)
37 \( 1 + (-0.370 + 0.928i)T \)
41 \( 1 + (0.267 - 0.963i)T \)
43 \( 1 + (0.947 + 0.319i)T \)
47 \( 1 + (-0.468 + 0.883i)T \)
53 \( 1 + (-0.796 - 0.605i)T \)
59 \( 1 + (-0.267 - 0.963i)T \)
61 \( 1 + (-0.907 - 0.419i)T \)
67 \( 1 + (0.976 - 0.214i)T \)
71 \( 1 + (-0.267 + 0.963i)T \)
73 \( 1 + (-0.725 + 0.687i)T \)
79 \( 1 + (-0.796 + 0.605i)T \)
83 \( 1 + (-0.561 + 0.827i)T \)
89 \( 1 + (0.994 + 0.108i)T \)
97 \( 1 + (0.561 + 0.827i)T \)
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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

−24.51749849896175302726011202529, −24.25558010486304501643284175174, −22.87879666546979259119740757150, −21.61705787362751817734251857313, −20.65060782543224285206207579957, −20.03895121711319858534783123825, −19.14315364402226914190671676382, −18.46746765382260320994445969674, −17.68198710491924606961684788274, −16.65503645883655592670400637572, −15.93452381896647529527193879546, −14.56744139251947711681823338705, −13.58129924725654674486121143567, −12.24625209830811656431649759949, −11.88082371021778027506694996950, −11.206850515366061683300365019907, −9.2902609127066605473724573410, −8.87836026145981253187284490484, −7.99930831246676020682866147124, −7.14350355373207733617669733387, −5.968476634573286033526146490025, −4.378617947963329330107605400464, −3.00829447676934084039595546929, −1.77428902359184083069559400783, −0.88095214658872181534344042861, 1.39859740063004532431974208281, 3.07193805629015641357317989353, 4.0576138867093730045490328129, 5.38025532614825292905324393854, 6.5818271427937812010187786204, 7.77560136506975256821816266762, 8.26827319690552955179395096887, 9.596281744843976690834080428109, 10.38586377493383250924600821876, 11.037072469823232192642157352581, 11.90500796902150013387756463347, 14.081673167248259798227007669449, 14.45945846096281214351909613503, 15.47021245179635343491218726588, 15.9872580400598459752040143085, 17.356305456485538887870769639022, 17.623722854247646753454386722433, 19.02954209783275277138131508272, 19.84965366033385034326414522172, 20.37843719399442623845144829686, 21.488674496692492693485391821250, 22.64691785307858594122100099533, 23.31024090976070845627350300616, 24.41305230466122017626856822736, 25.4999838436540778012112703625

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