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Properties

Label 2-231-7.6-c2-0-6
Degree $2$
Conductor $231$
Sign $0.0103 - 0.999i$
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  − 1.23·2-s + 1.73i·3-s − 2.48·4-s − 1.05i·5-s − 2.13i·6-s + (6.99 + 0.0726i)7-s + 7.98·8-s − 2.99·9-s + 1.29i·10-s − 3.31·11-s − 4.29i·12-s − 5.76i·13-s + (−8.62 − 0.0895i)14-s + 1.81·15-s + 0.0898·16-s + 28.6i·17-s + ⋯
L(s)  = 1  − 0.616·2-s + 0.577i·3-s − 0.620·4-s − 0.210i·5-s − 0.355i·6-s + (0.999 + 0.0103i)7-s + 0.998·8-s − 0.333·9-s + 0.129i·10-s − 0.301·11-s − 0.358i·12-s − 0.443i·13-s + (−0.615 − 0.00639i)14-s + 0.121·15-s + 0.00561·16-s + 1.68i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0103 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0103 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.0103 - 0.999i$
Analytic conductor: \(6.29429\)
Root analytic conductor: \(2.50884\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1),\ 0.0103 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.636247 + 0.629675i\)
\(L(\frac12)\) \(\approx\) \(0.636247 + 0.629675i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73iT \)
7 \( 1 + (-6.99 - 0.0726i)T \)
11 \( 1 + 3.31T \)
good2 \( 1 + 1.23T + 4T^{2} \)
5 \( 1 + 1.05iT - 25T^{2} \)
13 \( 1 + 5.76iT - 169T^{2} \)
17 \( 1 - 28.6iT - 289T^{2} \)
19 \( 1 - 28.5iT - 361T^{2} \)
23 \( 1 + 10.1T + 529T^{2} \)
29 \( 1 + 26.5T + 841T^{2} \)
31 \( 1 - 49.4iT - 961T^{2} \)
37 \( 1 + 3.18T + 1.36e3T^{2} \)
41 \( 1 - 35.4iT - 1.68e3T^{2} \)
43 \( 1 - 82.5T + 1.84e3T^{2} \)
47 \( 1 + 1.30iT - 2.20e3T^{2} \)
53 \( 1 + 34.1T + 2.80e3T^{2} \)
59 \( 1 - 53.5iT - 3.48e3T^{2} \)
61 \( 1 + 64.6iT - 3.72e3T^{2} \)
67 \( 1 + 3.87T + 4.48e3T^{2} \)
71 \( 1 + 14.4T + 5.04e3T^{2} \)
73 \( 1 + 41.8iT - 5.32e3T^{2} \)
79 \( 1 - 62.9T + 6.24e3T^{2} \)
83 \( 1 + 151. iT - 6.88e3T^{2} \)
89 \( 1 + 34.8iT - 7.92e3T^{2} \)
97 \( 1 - 110. iT - 9.40e3T^{2} \)
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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

−12.22442722484725870173172759551, −10.67052657346701470205600801988, −10.48369315005025032729670467664, −9.199306262804479208520922375430, −8.323437829037476513815606161739, −7.76820538610207026300185398414, −5.83247010096703140395231024552, −4.80942413155065756020602864103, −3.76342928250669788731430306281, −1.50829753078340562149852108207, 0.66541953609850205219701912510, 2.38024440752500907791695711198, 4.40524176606037494255933396563, 5.38729073650146899484699283312, 7.10664346875851925423665831557, 7.71967262469418917669882424857, 8.849390193900382592776701464699, 9.517453792818565057501894626513, 10.91795071356076976204210827549, 11.47593112835704866953467919284

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