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Properties

Label 2-403-403.295-c1-0-16
Degree $2$
Conductor $403$
Sign $0.984 + 0.172i$
Analytic cond. $3.21797$
Root an. cond. $1.79387$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.273 + 2.60i)2-s + (−2.04 + 0.909i)3-s + (−4.74 + 1.00i)4-s − 3·5-s + (−2.92 − 5.06i)6-s + (2.93 − 0.623i)7-s + (−2.30 − 7.10i)8-s + (1.33 − 1.48i)9-s + (−0.820 − 7.81i)10-s + (1.49 + 1.66i)11-s + (8.78 − 6.37i)12-s + (−3.54 + 0.632i)13-s + (2.42 + 7.46i)14-s + (6.12 − 2.72i)15-s + (9.00 − 4.00i)16-s + (−0.924 + 1.02i)17-s + ⋯
L(s)  = 1  + (0.193 + 1.84i)2-s + (−1.17 + 0.525i)3-s + (−2.37 + 0.504i)4-s − 1.34·5-s + (−1.19 − 2.06i)6-s + (1.10 − 0.235i)7-s + (−0.816 − 2.51i)8-s + (0.446 − 0.495i)9-s + (−0.259 − 2.47i)10-s + (0.451 + 0.501i)11-s + (2.53 − 1.84i)12-s + (−0.984 + 0.175i)13-s + (0.648 + 1.99i)14-s + (1.58 − 0.704i)15-s + (2.25 − 1.00i)16-s + (−0.224 + 0.249i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.984 + 0.172i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.984 + 0.172i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (3.54 - 0.632i)T \)
31 \( 1 + (-5.28 + 1.76i)T \)
good2 \( 1 + (-0.273 - 2.60i)T + (-1.95 + 0.415i)T^{2} \)
3 \( 1 + (2.04 - 0.909i)T + (2.00 - 2.22i)T^{2} \)
5 \( 1 + 3T + 5T^{2} \)
7 \( 1 + (-2.93 + 0.623i)T + (6.39 - 2.84i)T^{2} \)
11 \( 1 + (-1.49 - 1.66i)T + (-1.14 + 10.9i)T^{2} \)
17 \( 1 + (0.924 - 1.02i)T + (-1.77 - 16.9i)T^{2} \)
19 \( 1 + (5.48 + 2.44i)T + (12.7 + 14.1i)T^{2} \)
23 \( 1 + (-3.68 - 0.782i)T + (21.0 + 9.35i)T^{2} \)
29 \( 1 + (-0.587 - 5.58i)T + (-28.3 + 6.02i)T^{2} \)
37 \( 1 + (-4.04 + 7.00i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.765 + 7.28i)T + (-40.1 + 8.52i)T^{2} \)
43 \( 1 + (5.91 + 2.63i)T + (28.7 + 31.9i)T^{2} \)
47 \( 1 + (6.85 + 4.97i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.118 + 0.363i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (0.707 - 6.72i)T + (-57.7 - 12.2i)T^{2} \)
61 \( 1 + (5.78 + 10.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.11 - 10.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.02 - 3.36i)T + (-7.42 - 70.6i)T^{2} \)
73 \( 1 + (2.14 - 6.60i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (5.07 + 15.6i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (9.66 - 7.02i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (1.24 + 1.37i)T + (-9.30 + 88.5i)T^{2} \)
97 \( 1 + (0.604 - 0.128i)T + (88.6 - 39.4i)T^{2} \)
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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

−11.34302526094850835852844782761, −10.38209678148412004260505348573, −8.990331766803182429292826383230, −8.164886661727360429799017177986, −7.29035698144989965055586541252, −6.65067427451126147566843120372, −5.34062808333364824405762146828, −4.59265405928374448558763414562, −4.18331137187489442870791741087, 0, 1.36258020033197012993233889458, 2.96884146657611589151881090630, 4.44196001463483269870784846513, 4.86584692609589067645412999019, 6.34433146161441727938719392701, 7.87962650413921066229864887961, 8.596698749891205440296801823774, 9.948558321745008848836848249748, 11.00030794710999238638888560674

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