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Properties

Label 2-231-231.230-c2-0-30
Degree $2$
Conductor $231$
Sign $0.454 - 0.890i$
Analytic cond. $6.29429$
Root an. cond. $2.50884$
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  + 2.18·2-s + (−0.798 + 2.89i)3-s + 0.766·4-s + 9.25·5-s + (−1.74 + 6.31i)6-s + (3.46 + 6.08i)7-s − 7.05·8-s + (−7.72 − 4.62i)9-s + 20.2·10-s + (7.25 − 8.26i)11-s + (−0.612 + 2.21i)12-s − 3.97·13-s + (7.56 + 13.2i)14-s + (−7.39 + 26.7i)15-s − 18.4·16-s + 26.9i·17-s + ⋯
L(s)  = 1  + 1.09·2-s + (−0.266 + 0.963i)3-s + 0.191·4-s + 1.85·5-s + (−0.290 + 1.05i)6-s + (0.495 + 0.868i)7-s − 0.882·8-s + (−0.858 − 0.513i)9-s + 2.02·10-s + (0.659 − 0.751i)11-s + (−0.0510 + 0.184i)12-s − 0.306·13-s + (0.540 + 0.948i)14-s + (−0.492 + 1.78i)15-s − 1.15·16-s + 1.58i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.454 - 0.890i$
Analytic conductor: \(6.29429\)
Root analytic conductor: \(2.50884\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (230, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1),\ 0.454 - 0.890i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.50573 + 1.53468i\)
\(L(\frac12)\) \(\approx\) \(2.50573 + 1.53468i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.798 - 2.89i)T \)
7 \( 1 + (-3.46 - 6.08i)T \)
11 \( 1 + (-7.25 + 8.26i)T \)
good2 \( 1 - 2.18T + 4T^{2} \)
5 \( 1 - 9.25T + 25T^{2} \)
13 \( 1 + 3.97T + 169T^{2} \)
17 \( 1 - 26.9iT - 289T^{2} \)
19 \( 1 - 4.49T + 361T^{2} \)
23 \( 1 + 4.85iT - 529T^{2} \)
29 \( 1 + 29.2T + 841T^{2} \)
31 \( 1 + 42.0iT - 961T^{2} \)
37 \( 1 + 0.532T + 1.36e3T^{2} \)
41 \( 1 + 17.2iT - 1.68e3T^{2} \)
43 \( 1 + 59.2iT - 1.84e3T^{2} \)
47 \( 1 - 18.0T + 2.20e3T^{2} \)
53 \( 1 + 51.0iT - 2.80e3T^{2} \)
59 \( 1 - 59.2T + 3.48e3T^{2} \)
61 \( 1 + 72.8T + 3.72e3T^{2} \)
67 \( 1 + 87.6T + 4.48e3T^{2} \)
71 \( 1 + 32.3iT - 5.04e3T^{2} \)
73 \( 1 - 56.2T + 5.32e3T^{2} \)
79 \( 1 + 49.2iT - 6.24e3T^{2} \)
83 \( 1 + 44.3iT - 6.88e3T^{2} \)
89 \( 1 + 21.1T + 7.92e3T^{2} \)
97 \( 1 - 8.09iT - 9.40e3T^{2} \)
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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

−12.27785283838832944488060889874, −11.26869824311336674076778995720, −10.21699973018042849113908894514, −9.223211662660960800569016936260, −8.720019261134412500886546789198, −6.12069239480819896420920458363, −5.86559000709419034466079326872, −4.98968217452653297497704980788, −3.67435899514690861400259534524, −2.22425184688455027867675711993, 1.42245998089776168813554685927, 2.77013775930631778476631788335, 4.73306413365231757077038297654, 5.47136452065696439852868060677, 6.53639967899892481314983194205, 7.30204710057945146590047109678, 9.045360639834398431620693459338, 9.830108707534162555395661772221, 11.13908193947815257712700342878, 12.15950513686206042289037797928

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