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Properties

Label 2-403-403.347-c2-0-25
Degree $2$
Conductor $403$
Sign $-0.675 - 0.737i$
Analytic cond. $10.9809$
Root an. cond. $3.31375$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.47 + 2.55i)2-s + 1.72i·3-s + (−2.36 − 4.09i)4-s + (0.496 − 0.859i)5-s + (−4.40 − 2.54i)6-s + (5.78 − 10.0i)7-s + 2.14·8-s + 6.03·9-s + (1.46 + 2.53i)10-s + (−16.5 + 9.56i)11-s + (7.04 − 4.07i)12-s + (−2.52 − 12.7i)13-s + (17.1 + 29.6i)14-s + (1.48 + 0.854i)15-s + (6.28 − 10.8i)16-s + (25.7 + 14.8i)17-s + ⋯
L(s)  = 1  + (−0.738 + 1.27i)2-s + 0.574i·3-s + (−0.590 − 1.02i)4-s + (0.0992 − 0.171i)5-s + (−0.734 − 0.424i)6-s + (0.827 − 1.43i)7-s + 0.268·8-s + 0.670·9-s + (0.146 + 0.253i)10-s + (−1.50 + 0.869i)11-s + (0.587 − 0.339i)12-s + (−0.194 − 0.980i)13-s + (1.22 + 2.11i)14-s + (0.0986 + 0.0569i)15-s + (0.392 − 0.680i)16-s + (1.51 + 0.874i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.675 - 0.737i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.675 - 0.737i$
Analytic conductor: \(10.9809\)
Root analytic conductor: \(3.31375\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1),\ -0.675 - 0.737i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.469055 + 1.06623i\)
\(L(\frac12)\) \(\approx\) \(0.469055 + 1.06623i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (2.52 + 12.7i)T \)
31 \( 1 + (-1.42 - 30.9i)T \)
good2 \( 1 + (1.47 - 2.55i)T + (-2 - 3.46i)T^{2} \)
3 \( 1 - 1.72iT - 9T^{2} \)
5 \( 1 + (-0.496 + 0.859i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (-5.78 + 10.0i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (16.5 - 9.56i)T + (60.5 - 104. i)T^{2} \)
17 \( 1 + (-25.7 - 14.8i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (14.5 - 25.2i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-28.0 - 16.2i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (7.29 + 4.21i)T + (420.5 + 728. i)T^{2} \)
37 \( 1 + 24.1iT - 1.36e3T^{2} \)
41 \( 1 + (-12.3 - 21.3i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-3.73 - 2.15i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 - 78.9T + 2.20e3T^{2} \)
53 \( 1 + (36.1 + 20.8i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (5.33 - 9.23i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-97.9 + 56.5i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-21.0 - 36.5i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 8.06T + 5.04e3T^{2} \)
73 \( 1 + (40.1 + 23.1i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-30.1 + 17.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (45.9 + 26.5i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (11.4 + 6.60i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-49.7 - 86.1i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−10.63715237320820452181220461305, −10.39642444785847992728766999613, −9.626045108691391395252631522649, −8.236493701054435188124595389827, −7.62897505808232820038835490142, −7.18944921256990689601420027173, −5.55215108952305024055486863285, −4.92837459274625965432560298090, −3.59339258908009641893602340444, −1.20509063165954560765409789053, 0.77911485713022201218554495829, 2.30809144920433799290258629302, 2.74468036153254787015538619866, 4.73901182945142007413822329475, 5.84442736157922040488594429195, 7.23539555316611437740930948532, 8.321461395100943734324045637262, 8.922178046395805023702673052108, 9.899682082073926727383819328605, 10.88003580412202572785097902869

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