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Properties

Label 2-203-203.115-c2-0-32
Degree $2$
Conductor $203$
Sign $-0.841 + 0.540i$
Analytic cond. $5.53134$
Root an. cond. $2.35188$
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  + (−0.345 + 0.199i)2-s + (2.20 − 3.82i)3-s + (−1.92 + 3.32i)4-s + (−1.32 + 0.767i)5-s + 1.76i·6-s + (−6.23 − 3.18i)7-s − 3.12i·8-s + (−5.24 − 9.08i)9-s + (0.306 − 0.530i)10-s + (−12.5 − 7.25i)11-s + (8.47 + 14.6i)12-s − 15.5i·13-s + (2.79 − 0.142i)14-s + 6.77i·15-s + (−7.05 − 12.2i)16-s + (3.21 − 5.57i)17-s + ⋯
L(s)  = 1  + (−0.172 + 0.0997i)2-s + (0.735 − 1.27i)3-s + (−0.480 + 0.831i)4-s + (−0.265 + 0.153i)5-s + 0.293i·6-s + (−0.890 − 0.455i)7-s − 0.391i·8-s + (−0.582 − 1.00i)9-s + (0.0306 − 0.0530i)10-s + (−1.14 − 0.659i)11-s + (0.706 + 1.22i)12-s − 1.19i·13-s + (0.199 − 0.0101i)14-s + 0.451i·15-s + (−0.441 − 0.763i)16-s + (0.189 − 0.327i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 + 0.540i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(203\)    =    \(7 \cdot 29\)
Sign: $-0.841 + 0.540i$
Analytic conductor: \(5.53134\)
Root analytic conductor: \(2.35188\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{203} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 203,\ (\ :1),\ -0.841 + 0.540i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.216792 - 0.738659i\)
\(L(\frac12)\) \(\approx\) \(0.216792 - 0.738659i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (6.23 + 3.18i)T \)
29 \( 1 + (28.7 + 3.77i)T \)
good2 \( 1 + (0.345 - 0.199i)T + (2 - 3.46i)T^{2} \)
3 \( 1 + (-2.20 + 3.82i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (1.32 - 0.767i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (12.5 + 7.25i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 15.5iT - 169T^{2} \)
17 \( 1 + (-3.21 + 5.57i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-3.09 - 5.36i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (0.204 + 0.354i)T + (-264.5 + 458. i)T^{2} \)
31 \( 1 + (14.7 - 25.5i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-52.1 + 30.1i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 18.7T + 1.68e3T^{2} \)
43 \( 1 + 59.4iT - 1.84e3T^{2} \)
47 \( 1 + (-3.74 - 6.49i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-16.0 + 27.8i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (69.7 + 40.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-24.2 - 41.9i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (27.0 - 46.8i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 99.5T + 5.04e3T^{2} \)
73 \( 1 + (-12.9 + 22.4i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (68.7 - 39.6i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 13.4iT - 6.88e3T^{2} \)
89 \( 1 + (37.1 + 64.3i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 112.T + 9.40e3T^{2} \)
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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

−12.26737854924106080532905357572, −10.84685624155160815235222473255, −9.585321700069638267110019639696, −8.458918524678255264883277978394, −7.64844275216860229127003926818, −7.21923065782254461881011859230, −5.65745863336285322668284760770, −3.58251463872001611104976662073, −2.78439914243616404853083844387, −0.40975503857199821367238385935, 2.45557499421954223758325369528, 4.01838721920746438712883351470, 4.90108925757421709141921065961, 6.17181108306082814631876279164, 7.905765837661027001371169482089, 9.120969169465729028322956701740, 9.563097443385125184095977541493, 10.26734714217081374730820387252, 11.32525466913109509651650973006, 12.74833202385214965458487459257

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