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Properties

Label 2-403-13.9-c1-0-15
Degree $2$
Conductor $403$
Sign $-0.154 - 0.987i$
Analytic cond. $3.21797$
Root an. cond. $1.79387$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0205 − 0.0355i)2-s + (−1.04 + 1.81i)3-s + (0.999 + 1.73i)4-s + 1.98·5-s + (0.0430 + 0.0744i)6-s + (1.34 + 2.32i)7-s + 0.164·8-s + (−0.690 − 1.19i)9-s + (0.0408 − 0.0707i)10-s + (1.80 − 3.12i)11-s − 4.18·12-s + (3.58 + 0.359i)13-s + 0.110·14-s + (−2.08 + 3.60i)15-s + (−1.99 + 3.45i)16-s + (−3.50 − 6.06i)17-s + ⋯
L(s)  = 1  + (0.0145 − 0.0251i)2-s + (−0.604 + 1.04i)3-s + (0.499 + 0.865i)4-s + 0.889·5-s + (0.0175 + 0.0304i)6-s + (0.507 + 0.879i)7-s + 0.0580·8-s + (−0.230 − 0.398i)9-s + (0.0129 − 0.0223i)10-s + (0.544 − 0.942i)11-s − 1.20·12-s + (0.995 + 0.0997i)13-s + 0.0295·14-s + (−0.537 + 0.931i)15-s + (−0.498 + 0.863i)16-s + (−0.849 − 1.47i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.154 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.154 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.154 - 0.987i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.154 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02402 + 1.19706i\)
\(L(\frac12)\) \(\approx\) \(1.02402 + 1.19706i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-3.58 - 0.359i)T \)
31 \( 1 + T \)
good2 \( 1 + (-0.0205 + 0.0355i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.04 - 1.81i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 1.98T + 5T^{2} \)
7 \( 1 + (-1.34 - 2.32i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.80 + 3.12i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.50 + 6.06i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.0170 - 0.0294i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.38 - 2.39i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.00 + 5.20i)T + (-14.5 - 25.1i)T^{2} \)
37 \( 1 + (-0.429 + 0.744i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.0394 + 0.0683i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.629 - 1.08i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.99T + 47T^{2} \)
53 \( 1 + 3.93T + 53T^{2} \)
59 \( 1 + (-1.76 - 3.05i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.35 + 4.08i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.85 + 8.41i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.26 + 3.93i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 0.0228T + 73T^{2} \)
79 \( 1 - 12.1T + 79T^{2} \)
83 \( 1 - 5.82T + 83T^{2} \)
89 \( 1 + (-1.19 + 2.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.56 - 2.71i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.39817960708070641352481547961, −10.89594784234879765440967497260, −9.613683756026193989570233742023, −8.957017132132239616758160136139, −7.990882993886594421568873721202, −6.48740437759126255935955069015, −5.77181471304476030692050461891, −4.71263523830410094786378914534, −3.50110620307746812296679698350, −2.14815543939201557569519436442, 1.30127615171135681281841103475, 1.90041616853404503306649177445, 4.21520686181754301269483512563, 5.54684547461812127919829702390, 6.50488610696689483354732845396, 6.76460760877267992036675648541, 8.031114557161347479654779011203, 9.352162993881878160049311071601, 10.40594395629875874794832923820, 10.89208040735066360417894155170

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