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Properties

Label 1-403-403.150-r0-0-0
Degree $1$
Conductor $403$
Sign $-0.954 - 0.299i$
Analytic cond. $1.87152$
Root an. cond. $1.87152$
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  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + (−0.866 − 0.5i)5-s − i·6-s − i·7-s − i·8-s + (−0.5 − 0.866i)9-s − 10-s + i·11-s + (−0.5 − 0.866i)12-s + (−0.5 − 0.866i)14-s + (−0.866 + 0.5i)15-s + (−0.5 − 0.866i)16-s + 17-s + (−0.866 − 0.5i)18-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + (−0.866 − 0.5i)5-s − i·6-s − i·7-s − i·8-s + (−0.5 − 0.866i)9-s − 10-s + i·11-s + (−0.5 − 0.866i)12-s + (−0.5 − 0.866i)14-s + (−0.866 + 0.5i)15-s + (−0.5 − 0.866i)16-s + 17-s + (−0.866 − 0.5i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.954 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.954 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.954 - 0.299i$
Analytic conductor: \(1.87152\)
Root analytic conductor: \(1.87152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (150, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 403,\ (0:\ ),\ -0.954 - 0.299i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3148296128 - 2.052248475i\)
\(L(\frac12)\) \(\approx\) \(0.3148296128 - 2.052248475i\)
\(L(1)\) \(\approx\) \(1.094767198 - 1.267431016i\)
\(L(1)\) \(\approx\) \(1.094767198 - 1.267431016i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 - iT \)
11 \( 1 + iT \)
17 \( 1 + T \)
19 \( 1 + iT \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + iT \)
43 \( 1 + T \)
47 \( 1 - iT \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 - iT \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + iT \)
71 \( 1 + (0.866 - 0.5i)T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−24.64571653804410815291882799697, −23.9028276737467723879838440291, −22.85449788526693862571983333156, −22.10674596148947716864044994469, −21.512439331324601365507539026203, −20.74983046253867322378660321919, −19.54366419746973961647959209115, −18.95587839809494302226578419817, −17.52462453682737096704669390914, −16.32216228466621465890726294600, −15.700203171428224122964247962774, −15.201889852436625366428175799, −14.28568983028145706846591139872, −13.56856682222293595245544592725, −12.15053425568410366849007675762, −11.510969552525376001818144821034, −10.588832863916566469239844219385, −9.109551036238952273234796169953, −8.27522488940920425746155191709, −7.486834899547238003479357224609, −6.076819556727542000683632785344, −5.25330042867096082788820131712, −4.1178617681196901078963346877, −3.2327181999844730628066797087, −2.55819467188960616433055129026, 0.8889275347006580295224342465, 1.91239931006052167030432117736, 3.361769596560199566671070925916, 4.05193202656088930598608773722, 5.16978572239413654793378171904, 6.58888580971517953159145078161, 7.384286284725276631409168726737, 8.21965107377103073994218160993, 9.66510636681367263241475333662, 10.61829562487516778293997987359, 11.89095086180743439517330469535, 12.42158062850286625499205922605, 13.09439782927545490251360479714, 14.28885573090621267536162921530, 14.63998660941573577515517985545, 15.90471879711896521093967084165, 16.82414103134493968314207848960, 18.13350957705286040040758349680, 19.1004615365943029336083943691, 19.8199900401515223114438445569, 20.456103659880566557434703792093, 20.9538900410863191736620068894, 22.6226073315267346338926532653, 23.2527865707428108957657179196, 23.6896304187060552802353798087

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