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Properties

Label 1-407-407.19-r0-0-0
Degree $1$
Conductor $407$
Sign $0.261 - 0.965i$
Analytic cond. $1.89010$
Root an. cond. $1.89010$
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  + (−0.139 + 0.990i)2-s + (−0.438 − 0.898i)3-s + (−0.961 − 0.275i)4-s + (−0.927 − 0.374i)5-s + (0.951 − 0.309i)6-s + (0.241 + 0.970i)7-s + (0.406 − 0.913i)8-s + (−0.615 + 0.788i)9-s + (0.5 − 0.866i)10-s + (0.173 + 0.984i)12-s + (0.999 − 0.0348i)13-s + (−0.994 + 0.104i)14-s + (0.0697 + 0.997i)15-s + (0.848 + 0.529i)16-s + (−0.999 − 0.0348i)17-s + (−0.694 − 0.719i)18-s + ⋯
L(s)  = 1  + (−0.139 + 0.990i)2-s + (−0.438 − 0.898i)3-s + (−0.961 − 0.275i)4-s + (−0.927 − 0.374i)5-s + (0.951 − 0.309i)6-s + (0.241 + 0.970i)7-s + (0.406 − 0.913i)8-s + (−0.615 + 0.788i)9-s + (0.5 − 0.866i)10-s + (0.173 + 0.984i)12-s + (0.999 − 0.0348i)13-s + (−0.994 + 0.104i)14-s + (0.0697 + 0.997i)15-s + (0.848 + 0.529i)16-s + (−0.999 − 0.0348i)17-s + (−0.694 − 0.719i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.261 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.261 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $0.261 - 0.965i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ 0.261 - 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3662155355 - 0.2801992626i\)
\(L(\frac12)\) \(\approx\) \(0.3662155355 - 0.2801992626i\)
\(L(1)\) \(\approx\) \(0.6009046378 + 0.04008615327i\)
\(L(1)\) \(\approx\) \(0.6009046378 + 0.04008615327i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.139 + 0.990i)T \)
3 \( 1 + (-0.438 - 0.898i)T \)
5 \( 1 + (-0.927 - 0.374i)T \)
7 \( 1 + (0.241 + 0.970i)T \)
13 \( 1 + (0.999 - 0.0348i)T \)
17 \( 1 + (-0.999 - 0.0348i)T \)
19 \( 1 + (0.898 - 0.438i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + (-0.994 - 0.104i)T \)
31 \( 1 + (-0.951 - 0.309i)T \)
41 \( 1 + (0.559 - 0.829i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.104 - 0.994i)T \)
53 \( 1 + (-0.374 - 0.927i)T \)
59 \( 1 + (-0.0697 - 0.997i)T \)
61 \( 1 + (0.788 - 0.615i)T \)
67 \( 1 + (-0.766 - 0.642i)T \)
71 \( 1 + (0.990 - 0.139i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (-0.529 - 0.848i)T \)
83 \( 1 + (-0.0348 + 0.999i)T \)
89 \( 1 + (-0.642 - 0.766i)T \)
97 \( 1 + (-0.743 + 0.669i)T \)
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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

−24.02403548585250419250122954354, −23.308899562410405615276782718949, −22.64295683838030170268993393608, −21.95901425264797965917752242342, −20.86234114501476692530860089016, −20.22794103223549165740834699145, −19.64896180978568023656147053373, −18.315241363562325900872216825675, −17.76142989844408036251932182099, −16.565540219897249767034650848790, −15.91129126721885031468840162110, −14.71452616013593265010125832453, −13.8802896894839755296658938237, −12.7548860344722173819028648130, −11.44259749971623526570045720384, −11.22422611629536788846771540113, −10.38292527359440464701925637406, −9.41966058947965579130064371904, −8.35894755073600507789087209193, −7.34463063210614018135223196469, −5.82680576837242337369799450604, −4.45161154440981523489318064572, −3.93422020340823561517082999597, −3.105088695179766621133376488638, −1.24169887854819483862309365823, 0.34466879014043497596409211946, 1.88869040238919616048027067444, 3.70347922012982206426063973287, 4.9916870766169795518423990382, 5.76589640061122744727867703685, 6.76046967089876787994320405314, 7.698112694585951248426259314139, 8.47861128633116408341652916749, 9.15700836826930343009299357550, 10.92559241110027943351739549761, 11.7412931681687511997237919586, 12.70986973147348684551953098183, 13.45259073220544611904490481894, 14.54869940397785731589499486491, 15.63595684020668079940616284878, 16.086285477144015490252608612695, 17.15132609278891405936815522521, 18.2059112889169516766281403202, 18.5057066414986514722780007555, 19.5050120319386977383571711302, 20.47716295069319946332165357816, 22.13800928452028091287745587051, 22.538817712791323551261697436118, 23.65769171278786479371714683841, 24.1568489263372400174796242694

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