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Properties

Label 1-403-403.278-r0-0-0
Degree $1$
Conductor $403$
Sign $-0.957 - 0.289i$
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  − i·2-s − 3-s − 4-s − i·5-s + i·6-s + i·7-s + i·8-s + 9-s − 10-s − i·11-s + 12-s + 14-s + i·15-s + 16-s + 17-s − i·18-s + ⋯
L(s)  = 1  − i·2-s − 3-s − 4-s − i·5-s + i·6-s + i·7-s + i·8-s + 9-s − 10-s − i·11-s + 12-s + 14-s + i·15-s + 16-s + 17-s − i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.957 - 0.289i$
Analytic conductor: \(1.87152\)
Root analytic conductor: \(1.87152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (278, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 403,\ (0:\ ),\ -0.957 - 0.289i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1012528031 - 0.6838229577i\)
\(L(\frac12)\) \(\approx\) \(0.1012528031 - 0.6838229577i\)
\(L(1)\) \(\approx\) \(0.5241042358 - 0.4669310944i\)
\(L(1)\) \(\approx\) \(0.5241042358 - 0.4669310944i\)

Euler product

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

−24.724445817003609867162021282149, −23.52003979061117265323377266791, −23.11695885838251522235127164234, −22.629587456871983030315261899906, −21.66074374613152535308965007583, −20.55939345498912251349192738437, −19.00098150983091517428471425031, −18.48968059313459771834832567437, −17.46273309808327078680727058342, −16.99027462513987661356771589387, −16.10424235345290806265620653693, −15.030140153314561934318912527456, −14.4195011619579794583436498698, −13.30357876311932408314704995371, −12.413024912628590845449763498294, −11.19392059744044671973163806654, −10.15850854467873741190956969692, −9.73421243965031112367979039109, −7.8250274655449601447148632909, −7.22515808007924694826721263284, −6.5038335472757068225680125534, −5.50631398315045810877498728144, −4.44019932044032077057396136619, −3.495895786214120468193489800812, −1.33048626440042165210630588392, 0.53656517806838309558268311625, 1.66130977585249150463661410726, 3.10684956524393306673994838660, 4.41543562684047598817988973940, 5.37056642362441235448334442569, 5.86621125455799566726753304485, 7.69479331355980497746558203409, 8.94882865230525078270064982005, 9.40541753652437757629472675430, 10.794996754075358869472607871105, 11.43092661381018166065473623275, 12.3578995124874046071480703487, 12.81189953938063274058412901399, 13.8463053083356461989069110186, 15.26096649551305211788900472837, 16.2999801665827858475552542166, 17.07087792795031236275519594815, 17.920474229920446610432364682347, 18.88120406188498712308820792293, 19.41611705928719878485431731858, 20.88242860247361593117578072628, 21.2680177621401668211100942237, 22.080497025711381691093467299988, 22.893024539549357584281127007, 23.95240261573246392957395587381

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