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Properties

Label 1-407-407.344-r0-0-0
Degree $1$
Conductor $407$
Sign $0.250 + 0.968i$
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  + (−0.669 − 0.743i)2-s + (0.913 + 0.406i)3-s + (−0.104 + 0.994i)4-s + (−0.669 + 0.743i)5-s + (−0.309 − 0.951i)6-s + (−0.104 + 0.994i)7-s + (0.809 − 0.587i)8-s + (0.669 + 0.743i)9-s + 10-s + (−0.5 + 0.866i)12-s + (0.978 − 0.207i)13-s + (0.809 − 0.587i)14-s + (−0.913 + 0.406i)15-s + (−0.978 − 0.207i)16-s + (0.978 + 0.207i)17-s + (0.104 − 0.994i)18-s + ⋯
L(s)  = 1  + (−0.669 − 0.743i)2-s + (0.913 + 0.406i)3-s + (−0.104 + 0.994i)4-s + (−0.669 + 0.743i)5-s + (−0.309 − 0.951i)6-s + (−0.104 + 0.994i)7-s + (0.809 − 0.587i)8-s + (0.669 + 0.743i)9-s + 10-s + (−0.5 + 0.866i)12-s + (0.978 − 0.207i)13-s + (0.809 − 0.587i)14-s + (−0.913 + 0.406i)15-s + (−0.978 − 0.207i)16-s + (0.978 + 0.207i)17-s + (0.104 − 0.994i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.250 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.250 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $0.250 + 0.968i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (344, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ 0.250 + 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8403657622 + 0.6506164003i\)
\(L(\frac12)\) \(\approx\) \(0.8403657622 + 0.6506164003i\)
\(L(1)\) \(\approx\) \(0.9093862042 + 0.1940902960i\)
\(L(1)\) \(\approx\) \(0.9093862042 + 0.1940902960i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.669 - 0.743i)T \)
3 \( 1 + (0.913 + 0.406i)T \)
5 \( 1 + (-0.669 + 0.743i)T \)
7 \( 1 + (-0.104 + 0.994i)T \)
13 \( 1 + (0.978 - 0.207i)T \)
17 \( 1 + (0.978 + 0.207i)T \)
19 \( 1 + (-0.913 - 0.406i)T \)
23 \( 1 - T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (0.913 + 0.406i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (0.669 + 0.743i)T \)
59 \( 1 + (-0.913 + 0.406i)T \)
61 \( 1 + (-0.669 + 0.743i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.669 - 0.743i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (0.978 - 0.207i)T \)
83 \( 1 + (-0.978 - 0.207i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.309 + 0.951i)T \)
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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

−24.2444799846742204864803072550, −23.44104942874535594083366452896, −23.10799749075955372429814818203, −21.09582940958252387168307522721, −20.40464726509477664906964796028, −19.66735941346792049525589907494, −18.993480362788015328187085584828, −18.129643671465945010665908413137, −16.97444257647241844881501175605, −16.28257149095520809272133864923, −15.47722002211460358504949846662, −14.46228670439852989343285023687, −13.70502503385245273704076923591, −12.85321112751965654510241215262, −11.58432704414015233378699258639, −10.30342675276413762836626107162, −9.47248524471547270998130096877, −8.32369835263425319551201404205, −8.001168126095634737040803374385, −7.01524762057894627111619433712, −5.99035762472459626031222112890, −4.429341226743624627595620974807, −3.6575087868562088743137582519, −1.77494279870810078445396159326, −0.73442570156652883207208994993, 1.70197424552872889418401956516, 2.87887705212240883681760273082, 3.42719402017363453195737280086, 4.536739082354219741876512865, 6.311696063196974129183808216977, 7.63192486322296313192165990260, 8.36761272297304071521700435863, 9.042341527435626863818746020725, 10.20592857840241155001924249839, 10.82037164279237323049985366174, 11.9311851536044690302760570901, 12.74752298900662052517682425042, 13.92135421670769258640290275317, 14.90562708281499135812093845405, 15.76541848519390209754079277435, 16.42537836078833368885207137308, 18.087512012199379154967033489086, 18.4921103205737170342861039362, 19.46842888727315092024369330768, 19.84838713088197421038920219414, 21.15022724864039236324145577392, 21.53854543704719161876493016804, 22.46612718155339248414101928546, 23.5051013196477367557237370062, 24.94599355456064209229483976739

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