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Properties

Label 2-403-31.25-c1-0-9
Degree $2$
Conductor $403$
Sign $0.968 + 0.250i$
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  − 2.54·2-s + (−0.358 − 0.621i)3-s + 4.46·4-s + (−1.54 + 2.68i)5-s + (0.911 + 1.57i)6-s + (−1.09 − 1.89i)7-s − 6.25·8-s + (1.24 − 2.15i)9-s + (3.93 − 6.81i)10-s + (−1.10 + 1.92i)11-s + (−1.59 − 2.77i)12-s + (0.5 − 0.866i)13-s + (2.78 + 4.81i)14-s + 2.22·15-s + 6.97·16-s + (−0.161 − 0.279i)17-s + ⋯
L(s)  = 1  − 1.79·2-s + (−0.207 − 0.358i)3-s + 2.23·4-s + (−0.692 + 1.19i)5-s + (0.372 + 0.644i)6-s + (−0.413 − 0.716i)7-s − 2.21·8-s + (0.414 − 0.717i)9-s + (1.24 − 2.15i)10-s + (−0.334 + 0.579i)11-s + (−0.461 − 0.799i)12-s + (0.138 − 0.240i)13-s + (0.743 + 1.28i)14-s + 0.573·15-s + 1.74·16-s + (−0.0391 − 0.0677i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.968 + 0.250i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.444641 - 0.0566128i\)
\(L(\frac12)\) \(\approx\) \(0.444641 - 0.0566128i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-5.14 + 2.13i)T \)
good2 \( 1 + 2.54T + 2T^{2} \)
3 \( 1 + (0.358 + 0.621i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.54 - 2.68i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.09 + 1.89i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.10 - 1.92i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.161 + 0.279i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.35 - 2.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 1.21T + 23T^{2} \)
29 \( 1 - 8.29T + 29T^{2} \)
37 \( 1 + (0.391 + 0.678i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.98 + 8.64i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.01 - 8.68i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 4.94T + 47T^{2} \)
53 \( 1 + (-4.55 + 7.89i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.48 - 9.50i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 4.67T + 61T^{2} \)
67 \( 1 + (-6.07 + 10.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.25 - 3.91i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.38 + 7.60i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.33 - 7.51i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.415 - 0.720i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 16.0T + 89T^{2} \)
97 \( 1 + 3.88T + 97T^{2} \)
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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

−10.88257680870470434112940195364, −10.20003641378166882602319990447, −9.662888172124221507374614636013, −8.332380168632823881242544367750, −7.44605730742125331435940552526, −6.99524259429920632962996630048, −6.22391804954537084231403848307, −3.87382360080335963669925713235, −2.58346136191852353864869778138, −0.78498172217564949035419489390, 0.902796118119267600595131297470, 2.59959454781665180654008999883, 4.46868733481509553103164009094, 5.69395042180354698103262516391, 6.97506058805671357564848677411, 8.108326138468474581401380558830, 8.532491169190046804426988660247, 9.356989658852633626806010679558, 10.18926028461331370237662956013, 11.06737234051700763909188928105

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