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Properties

Label 1-407-407.17-r0-0-0
Degree $1$
Conductor $407$
Sign $-0.954 + 0.298i$
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  + (0.829 + 0.559i)2-s + (−0.0348 + 0.999i)3-s + (0.374 + 0.927i)4-s + (0.898 + 0.438i)5-s + (−0.587 + 0.809i)6-s + (−0.990 − 0.139i)7-s + (−0.207 + 0.978i)8-s + (−0.997 − 0.0697i)9-s + (0.5 + 0.866i)10-s + (−0.939 + 0.342i)12-s + (0.970 + 0.241i)13-s + (−0.743 − 0.669i)14-s + (−0.469 + 0.882i)15-s + (−0.719 + 0.694i)16-s + (−0.970 + 0.241i)17-s + (−0.788 − 0.615i)18-s + ⋯
L(s)  = 1  + (0.829 + 0.559i)2-s + (−0.0348 + 0.999i)3-s + (0.374 + 0.927i)4-s + (0.898 + 0.438i)5-s + (−0.587 + 0.809i)6-s + (−0.990 − 0.139i)7-s + (−0.207 + 0.978i)8-s + (−0.997 − 0.0697i)9-s + (0.5 + 0.866i)10-s + (−0.939 + 0.342i)12-s + (0.970 + 0.241i)13-s + (−0.743 − 0.669i)14-s + (−0.469 + 0.882i)15-s + (−0.719 + 0.694i)16-s + (−0.970 + 0.241i)17-s + (−0.788 − 0.615i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.954 + 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.954 + 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $-0.954 + 0.298i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ -0.954 + 0.298i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2989604175 + 1.955910373i\)
\(L(\frac12)\) \(\approx\) \(0.2989604175 + 1.955910373i\)
\(L(1)\) \(\approx\) \(1.044627395 + 1.216967788i\)
\(L(1)\) \(\approx\) \(1.044627395 + 1.216967788i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.829 + 0.559i)T \)
3 \( 1 + (-0.0348 + 0.999i)T \)
5 \( 1 + (0.898 + 0.438i)T \)
7 \( 1 + (-0.990 - 0.139i)T \)
13 \( 1 + (0.970 + 0.241i)T \)
17 \( 1 + (-0.970 + 0.241i)T \)
19 \( 1 + (-0.999 - 0.0348i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (-0.743 + 0.669i)T \)
31 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (0.848 + 0.529i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.669 - 0.743i)T \)
53 \( 1 + (0.438 + 0.898i)T \)
59 \( 1 + (0.469 - 0.882i)T \)
61 \( 1 + (-0.0697 - 0.997i)T \)
67 \( 1 + (-0.173 - 0.984i)T \)
71 \( 1 + (0.559 + 0.829i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.694 + 0.719i)T \)
83 \( 1 + (0.241 + 0.970i)T \)
89 \( 1 + (-0.984 - 0.173i)T \)
97 \( 1 + (-0.406 + 0.913i)T \)
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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

−23.9104803760222874110054697447, −22.880220049699469826461771645313, −22.457761016559139120817161563436, −21.26095248922081569385967869837, −20.568638449679192680511375933355, −19.579915206799007240428124952412, −18.93966723982909501610473878275, −18.00582694001845161169329214626, −17.01827864981786305014939373141, −15.92650252901087069290388790018, −14.87055428492863683993174060191, −13.66685937394917259055246708419, −13.155913457285933716321441459907, −12.78226142435630788038123009699, −11.58864712582216692868948884110, −10.69009513632604909069233633731, −9.5017887468810369362298852228, −8.686447645918312330855224174498, −7.03893160829315257616401300086, −6.13549503281622704336899406582, −5.70757383350672746015923839194, −4.24405742741748918304961495482, −2.87445730489527902250566826272, −2.08445572338956269363942600221, −0.864659379782757683585594872163, 2.34222958584602585997511053135, 3.32398609286108798025648316852, 4.19642706340413763375700799595, 5.3454319650713937801062483792, 6.29855807435375572707001623293, 6.80803689285197305306878160302, 8.59749469341360524628351092999, 9.241996666600767048528011092033, 10.570623707516041853179497895127, 11.060285010408106665990008128045, 12.580298949972033445009541590220, 13.3929027854192473719280748113, 14.150446849862195757821583055163, 15.121137336709852095673228394938, 15.771098741122372820150450299493, 16.7367601316637764879499569392, 17.2610427492500863953295846116, 18.47305453088162251483954443064, 19.8164621672501086745679367378, 20.78174984790699821951133839606, 21.48726145721117039713347849658, 22.14360516993664172964873333650, 22.848226193362537463515226698962, 23.53898827399353020967172919041, 24.9562726681984455426554811218

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