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Properties

Label 1-349-349.85-r0-0-0
Degree $1$
Conductor $349$
Sign $0.983 - 0.179i$
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  + (−0.619 − 0.785i)2-s + (−0.0901 − 0.995i)3-s + (−0.232 + 0.972i)4-s + (0.750 + 0.661i)5-s + (−0.725 + 0.687i)6-s + (0.700 + 0.713i)7-s + (0.907 − 0.419i)8-s + (−0.983 + 0.179i)9-s + (0.0541 − 0.998i)10-s + (−0.994 − 0.108i)11-s + (0.989 + 0.143i)12-s + (−0.922 + 0.386i)13-s + (0.126 − 0.992i)14-s + (0.590 − 0.806i)15-s + (−0.891 − 0.452i)16-s + (0.907 + 0.419i)17-s + ⋯
L(s)  = 1  + (−0.619 − 0.785i)2-s + (−0.0901 − 0.995i)3-s + (−0.232 + 0.972i)4-s + (0.750 + 0.661i)5-s + (−0.725 + 0.687i)6-s + (0.700 + 0.713i)7-s + (0.907 − 0.419i)8-s + (−0.983 + 0.179i)9-s + (0.0541 − 0.998i)10-s + (−0.994 − 0.108i)11-s + (0.989 + 0.143i)12-s + (−0.922 + 0.386i)13-s + (0.126 − 0.992i)14-s + (0.590 − 0.806i)15-s + (−0.891 − 0.452i)16-s + (0.907 + 0.419i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(349\)
Sign: $0.983 - 0.179i$
Analytic conductor: \(1.62074\)
Root analytic conductor: \(1.62074\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 349,\ (0:\ ),\ 0.983 - 0.179i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9207140565 - 0.08334892977i\)
\(L(\frac12)\) \(\approx\) \(0.9207140565 - 0.08334892977i\)
\(L(1)\) \(\approx\) \(0.8033886430 - 0.2265058849i\)
\(L(1)\) \(\approx\) \(0.8033886430 - 0.2265058849i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 \)
good2 \( 1 + (-0.619 - 0.785i)T \)
3 \( 1 + (-0.0901 - 0.995i)T \)
5 \( 1 + (0.750 + 0.661i)T \)
7 \( 1 + (0.700 + 0.713i)T \)
11 \( 1 + (-0.994 - 0.108i)T \)
13 \( 1 + (-0.922 + 0.386i)T \)
17 \( 1 + (0.907 + 0.419i)T \)
19 \( 1 + (0.997 + 0.0721i)T \)
23 \( 1 + (-0.922 + 0.386i)T \)
29 \( 1 + (0.989 - 0.143i)T \)
31 \( 1 + (-0.856 + 0.515i)T \)
37 \( 1 + (0.796 + 0.605i)T \)
41 \( 1 + (0.907 - 0.419i)T \)
43 \( 1 + (0.403 + 0.915i)T \)
47 \( 1 + (-0.161 + 0.986i)T \)
53 \( 1 + (0.976 + 0.214i)T \)
59 \( 1 + (-0.0901 - 0.995i)T \)
61 \( 1 + (-0.370 - 0.928i)T \)
67 \( 1 + (-0.561 - 0.827i)T \)
71 \( 1 + (-0.0901 + 0.995i)T \)
73 \( 1 + (-0.968 + 0.250i)T \)
79 \( 1 + (0.976 - 0.214i)T \)
83 \( 1 + (0.197 - 0.980i)T \)
89 \( 1 + (0.530 - 0.847i)T \)
97 \( 1 + (0.750 - 0.661i)T \)
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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

−24.96962002357272708496734357992, −24.10024586355415406965603458939, −23.29810957054908337997166534866, −22.27470388387726203364857901634, −21.20101369616333569371761624331, −20.41859815216850681307920815647, −19.80554016206253448097701512501, −18.04833687452342805883609867467, −17.75189710446992367121192323520, −16.50145116754160870638041906726, −16.37559868654718092098276011897, −15.06450732800668046683939887667, −14.28063194687219965613097307362, −13.519460402122816123879963339409, −11.98186279129744440909623946728, −10.565883254064870890114486668332, −10.08914184511913358506947763407, −9.29075769400061020811207336136, −8.15433498788395137284672172603, −7.40137233644359212653381345069, −5.70076860886156656580498365208, −5.213856128617637083478333557371, −4.32947289324762840458424619510, −2.43355818481180454972690630335, −0.7603307159415483122206392946, 1.41320436785362308701744936555, 2.33296150446648183617943812802, 3.043117325576308039029371780042, 5.04698691597444661378281595340, 6.058839618198614114239338545247, 7.48970043745102428769749332602, 7.94216367296324059303279594925, 9.22769636750414426056562634142, 10.175705461681545156739036289558, 11.18736378291968492496564625017, 12.02291710889459182106399143889, 12.7763377071455751929204640276, 13.8845648958792666636904434920, 14.53779968574782775408294541894, 16.13639157028661207657729847503, 17.36517000163247866901777127248, 17.93306893949622260513671285987, 18.531390879059874967294824957492, 19.22183390599180344685981159285, 20.29058677268907235237819104223, 21.386727723814208427629386260616, 21.85803966426253884234700237618, 22.94057589024699151420541716513, 24.06455986016104813238061782312, 24.9740816168749061453265023323

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