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Properties

Label 1-407-407.26-r0-0-0
Degree $1$
Conductor $407$
Sign $-0.389 + 0.920i$
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  + (−0.978 − 0.207i)2-s + (−0.104 + 0.994i)3-s + (0.913 + 0.406i)4-s + (−0.978 + 0.207i)5-s + (0.309 − 0.951i)6-s + (0.913 + 0.406i)7-s + (−0.809 − 0.587i)8-s + (−0.978 − 0.207i)9-s + 10-s + (−0.5 + 0.866i)12-s + (0.669 − 0.743i)13-s + (−0.809 − 0.587i)14-s + (−0.104 − 0.994i)15-s + (0.669 + 0.743i)16-s + (0.669 + 0.743i)17-s + (0.913 + 0.406i)18-s + ⋯
L(s)  = 1  + (−0.978 − 0.207i)2-s + (−0.104 + 0.994i)3-s + (0.913 + 0.406i)4-s + (−0.978 + 0.207i)5-s + (0.309 − 0.951i)6-s + (0.913 + 0.406i)7-s + (−0.809 − 0.587i)8-s + (−0.978 − 0.207i)9-s + 10-s + (−0.5 + 0.866i)12-s + (0.669 − 0.743i)13-s + (−0.809 − 0.587i)14-s + (−0.104 − 0.994i)15-s + (0.669 + 0.743i)16-s + (0.669 + 0.743i)17-s + (0.913 + 0.406i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.389 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.389 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $-0.389 + 0.920i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ -0.389 + 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3731061255 + 0.5629950721i\)
\(L(\frac12)\) \(\approx\) \(0.3731061255 + 0.5629950721i\)
\(L(1)\) \(\approx\) \(0.5827732031 + 0.2720938832i\)
\(L(1)\) \(\approx\) \(0.5827732031 + 0.2720938832i\)

Euler product

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

−24.09140945820371096061304141996, −23.614403586784613616624789876504, −22.73773045480880286987358293202, −20.89822276656240198810849663185, −20.47867717379445462068429596658, −19.31210254561693193260521503798, −18.91772584402185339584141852110, −18.00957149608759311503518283837, −17.142678312049002511472311874967, −16.45564187701275031871829838618, −15.36047176901302778228772717128, −14.45594171772503145691737121855, −13.42144181686821799709505883009, −12.11761415962764323204170147696, −11.348007007058133699732046006028, −10.960553827935169023309369699065, −9.225374179883415896008953225404, −8.46190606327099398371235179553, −7.5106487894335842216508273455, −7.14819648325799484855164061566, −5.857858847536432781336547663527, −4.581722796373159836641127815683, −2.9544846390551698892752855702, −1.61396834638532653208302258013, −0.632228831370346014782211897605, 1.33532277307317509316986357262, 3.025456797986356711489091780425, 3.74818992080374076156072417710, 5.09689759543005129763012110051, 6.24317258328899997963971800090, 7.76918580959647757067349868761, 8.29037684296413558929431938586, 9.1522233996336976720486297580, 10.4661530730972962076473307544, 10.893961815271253870141816927943, 11.74639629842373985896273708890, 12.6011645567625468627710531538, 14.60188410978156043430525012495, 15.05206725429630966445971106654, 15.93455746285312378185933046661, 16.67402464913578297191114011102, 17.62055422531215631314292877418, 18.51656729797862421318674928522, 19.35791280839657079105674112941, 20.37101890377960960430896585832, 20.91801317402599415526094160883, 21.71025018767809255383182246567, 22.86521373381310215008424948710, 23.66574164683014973162051032639, 24.87141385863915964064930646154

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