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Properties

Label 2-253-11.10-c2-0-1
Degree $2$
Conductor $253$
Sign $0.367 - 0.930i$
Analytic cond. $6.89375$
Root an. cond. $2.62559$
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  − 2.64i·2-s − 0.454·3-s − 2.99·4-s − 5.27·5-s + 1.20i·6-s + 3.36i·7-s − 2.66i·8-s − 8.79·9-s + 13.9i·10-s + (4.03 − 10.2i)11-s + 1.36·12-s + 22.4i·13-s + 8.91·14-s + 2.40·15-s − 19.0·16-s + 19.8i·17-s + ⋯
L(s)  = 1  − 1.32i·2-s − 0.151·3-s − 0.748·4-s − 1.05·5-s + 0.200i·6-s + 0.481i·7-s − 0.332i·8-s − 0.977·9-s + 1.39i·10-s + (0.367 − 0.930i)11-s + 0.113·12-s + 1.72i·13-s + 0.636·14-s + 0.160·15-s − 1.18·16-s + 1.16i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(253\)    =    \(11 \cdot 23\)
Sign: $0.367 - 0.930i$
Analytic conductor: \(6.89375\)
Root analytic conductor: \(2.62559\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{253} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 253,\ (\ :1),\ 0.367 - 0.930i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.191207 + 0.130098i\)
\(L(\frac12)\) \(\approx\) \(0.191207 + 0.130098i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-4.03 + 10.2i)T \)
23 \( 1 - 4.79T \)
good2 \( 1 + 2.64iT - 4T^{2} \)
3 \( 1 + 0.454T + 9T^{2} \)
5 \( 1 + 5.27T + 25T^{2} \)
7 \( 1 - 3.36iT - 49T^{2} \)
13 \( 1 - 22.4iT - 169T^{2} \)
17 \( 1 - 19.8iT - 289T^{2} \)
19 \( 1 - 10.9iT - 361T^{2} \)
29 \( 1 - 7.02iT - 841T^{2} \)
31 \( 1 + 27.2T + 961T^{2} \)
37 \( 1 + 62.0T + 1.36e3T^{2} \)
41 \( 1 + 20.1iT - 1.68e3T^{2} \)
43 \( 1 + 79.7iT - 1.84e3T^{2} \)
47 \( 1 - 8.66T + 2.20e3T^{2} \)
53 \( 1 + 67.8T + 2.80e3T^{2} \)
59 \( 1 + 8.14T + 3.48e3T^{2} \)
61 \( 1 - 81.2iT - 3.72e3T^{2} \)
67 \( 1 + 37.3T + 4.48e3T^{2} \)
71 \( 1 + 48.1T + 5.04e3T^{2} \)
73 \( 1 - 34.8iT - 5.32e3T^{2} \)
79 \( 1 + 75.0iT - 6.24e3T^{2} \)
83 \( 1 + 22.7iT - 6.88e3T^{2} \)
89 \( 1 + 172.T + 7.92e3T^{2} \)
97 \( 1 - 27.8T + 9.40e3T^{2} \)
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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

−11.92488442871395929863990303464, −11.23053177018443805635634121322, −10.52950848067412702580533391336, −9.017545834817303517389160982308, −8.589169554554148185250181044061, −7.02219219277105303525161659986, −5.79731307760037422689003107261, −4.10448983061821665372530520270, −3.38046223666333241935952140782, −1.82611014602338636796435838536, 0.11795954656151824315955596769, 3.06620617633488614041780188018, 4.65282781524549288471765379128, 5.54825301202397630034904586577, 6.83599277033581153012671439327, 7.60223269004010601573947338681, 8.245881480368499940815134725680, 9.410223384601738040512109897134, 10.84459318543569494234871630911, 11.58225344292823292053885365196

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