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Properties

Label 1-3407-3407.1082-r0-0-0
Degree $1$
Conductor $3407$
Sign $0.911 - 0.411i$
Analytic cond. $15.8220$
Root an. cond. $15.8220$
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  + (0.739 + 0.672i)2-s + (−0.104 − 0.994i)3-s + (0.0948 + 0.995i)4-s + (0.282 − 0.959i)5-s + (0.592 − 0.805i)6-s + (0.997 + 0.0700i)7-s + (−0.599 + 0.800i)8-s + (−0.978 + 0.206i)9-s + (0.854 − 0.519i)10-s + (0.878 − 0.477i)11-s + (0.980 − 0.197i)12-s + (−0.861 − 0.508i)13-s + (0.690 + 0.722i)14-s + (−0.983 − 0.181i)15-s + (−0.982 + 0.188i)16-s + (0.911 + 0.411i)17-s + ⋯
L(s)  = 1  + (0.739 + 0.672i)2-s + (−0.104 − 0.994i)3-s + (0.0948 + 0.995i)4-s + (0.282 − 0.959i)5-s + (0.592 − 0.805i)6-s + (0.997 + 0.0700i)7-s + (−0.599 + 0.800i)8-s + (−0.978 + 0.206i)9-s + (0.854 − 0.519i)10-s + (0.878 − 0.477i)11-s + (0.980 − 0.197i)12-s + (−0.861 − 0.508i)13-s + (0.690 + 0.722i)14-s + (−0.983 − 0.181i)15-s + (−0.982 + 0.188i)16-s + (0.911 + 0.411i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3407\)
Sign: $0.911 - 0.411i$
Analytic conductor: \(15.8220\)
Root analytic conductor: \(15.8220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3407} (1082, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3407,\ (0:\ ),\ 0.911 - 0.411i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.185275154 - 0.6855742742i\)
\(L(\frac12)\) \(\approx\) \(3.185275154 - 0.6855742742i\)
\(L(1)\) \(\approx\) \(1.835020449 - 0.09329869582i\)
\(L(1)\) \(\approx\) \(1.835020449 - 0.09329869582i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3407 \( 1 \)
good2 \( 1 + (0.739 + 0.672i)T \)
3 \( 1 + (-0.104 - 0.994i)T \)
5 \( 1 + (0.282 - 0.959i)T \)
7 \( 1 + (0.997 + 0.0700i)T \)
11 \( 1 + (0.878 - 0.477i)T \)
13 \( 1 + (-0.861 - 0.508i)T \)
17 \( 1 + (0.911 + 0.411i)T \)
19 \( 1 + (0.852 + 0.522i)T \)
23 \( 1 + (0.944 - 0.329i)T \)
29 \( 1 + (0.885 + 0.464i)T \)
31 \( 1 + (0.807 - 0.589i)T \)
37 \( 1 + (-0.947 - 0.320i)T \)
41 \( 1 + (-0.897 - 0.440i)T \)
43 \( 1 + (-0.765 + 0.643i)T \)
47 \( 1 + (0.696 + 0.717i)T \)
53 \( 1 + (0.450 - 0.892i)T \)
59 \( 1 + (-0.507 + 0.861i)T \)
61 \( 1 + (0.796 + 0.604i)T \)
67 \( 1 + (0.761 + 0.647i)T \)
71 \( 1 + (0.290 - 0.957i)T \)
73 \( 1 + (0.965 - 0.258i)T \)
79 \( 1 + (-0.988 - 0.152i)T \)
83 \( 1 + (-0.523 - 0.852i)T \)
89 \( 1 + (-0.755 + 0.654i)T \)
97 \( 1 + (0.527 + 0.849i)T \)
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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

−18.94559157870614292580523639021, −18.285129165825817977253036778505, −17.24567476472637566957646783365, −17.081445116871934104313836164889, −15.63387092805296293826529486424, −15.287664408541953483724278333774, −14.5209672162382681815375589980, −14.10458214230381036379121209260, −13.71350169608781127779691183020, −12.171137921873077306287797711006, −11.77829991078235750969499461160, −11.28331200722280432542554253456, −10.46413647754096094741104362033, −9.86397314945402249977404294276, −9.428087580006830508523676795286, −8.41018469310500990747241259493, −7.09450699531258007466475401062, −6.7157905090937192275290258140, −5.48733983006617176114762981291, −5.05173900029636694394270062224, −4.38227224464683343212161568938, −3.48929851203534556402410842341, −2.884573460190214345146609599410, −2.04126578828726046345105140886, −1.05316015085452733720513004826, 0.8844407787178956711954294275, 1.574963045629820225842956785580, 2.59230576187751339818921839449, 3.47946316021826999987507096394, 4.546042349707147328476523485570, 5.31117998117414972239286052901, 5.61620789884376611580767501012, 6.57258881801323189000877873425, 7.32714002191675363166037101056, 8.10809117846524537407265962762, 8.43539571767611406311258789981, 9.2563744527002065888041297800, 10.44571550722310732948389054141, 11.64552197412028110420944894777, 11.96274180050326593957807032243, 12.497292239781754047488767816517, 13.2631562979751652840847520999, 13.99365065147552041676085947948, 14.395105133823019727962002745892, 15.07542498084629981324486106815, 16.13253040901979859265230316883, 16.93056205517012291605962550120, 17.17430025374152853911867797221, 17.7703406766334705011566953748, 18.61356109468639736401461271330

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