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Properties

Label 2-253-11.10-c2-0-41
Degree $2$
Conductor $253$
Sign $-0.910 + 0.413i$
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  − 1.15i·2-s + 2.03·3-s + 2.66·4-s − 8.63·5-s − 2.34i·6-s − 11.4i·7-s − 7.70i·8-s − 4.87·9-s + 9.97i·10-s + (−10.0 + 4.54i)11-s + 5.41·12-s + 13.8i·13-s − 13.2·14-s − 17.5·15-s + 1.75·16-s − 15.2i·17-s + ⋯
L(s)  = 1  − 0.577i·2-s + 0.677·3-s + 0.666·4-s − 1.72·5-s − 0.391i·6-s − 1.64i·7-s − 0.962i·8-s − 0.541·9-s + 0.997i·10-s + (−0.910 + 0.413i)11-s + 0.451·12-s + 1.06i·13-s − 0.948·14-s − 1.16·15-s + 0.109·16-s − 0.896i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 + 0.413i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(253\)    =    \(11 \cdot 23\)
Sign: $-0.910 + 0.413i$
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: \((2,\ 253,\ (\ :1),\ -0.910 + 0.413i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.248156 - 1.14679i\)
\(L(\frac12)\) \(\approx\) \(0.248156 - 1.14679i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (10.0 - 4.54i)T \)
23 \( 1 - 4.79T \)
good2 \( 1 + 1.15iT - 4T^{2} \)
3 \( 1 - 2.03T + 9T^{2} \)
5 \( 1 + 8.63T + 25T^{2} \)
7 \( 1 + 11.4iT - 49T^{2} \)
13 \( 1 - 13.8iT - 169T^{2} \)
17 \( 1 + 15.2iT - 289T^{2} \)
19 \( 1 + 20.7iT - 361T^{2} \)
29 \( 1 + 3.94iT - 841T^{2} \)
31 \( 1 - 23.2T + 961T^{2} \)
37 \( 1 - 8.15T + 1.36e3T^{2} \)
41 \( 1 + 57.8iT - 1.68e3T^{2} \)
43 \( 1 - 26.7iT - 1.84e3T^{2} \)
47 \( 1 - 33.6T + 2.20e3T^{2} \)
53 \( 1 + 36.1T + 2.80e3T^{2} \)
59 \( 1 - 75.7T + 3.48e3T^{2} \)
61 \( 1 + 8.02iT - 3.72e3T^{2} \)
67 \( 1 - 87.9T + 4.48e3T^{2} \)
71 \( 1 + 92.6T + 5.04e3T^{2} \)
73 \( 1 - 29.3iT - 5.32e3T^{2} \)
79 \( 1 + 59.4iT - 6.24e3T^{2} \)
83 \( 1 - 138. iT - 6.88e3T^{2} \)
89 \( 1 - 103.T + 7.92e3T^{2} \)
97 \( 1 + 0.294T + 9.40e3T^{2} \)
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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

−11.28996867588087863485119028505, −10.81858332346539464171007040902, −9.611114972535137210753534614827, −8.285719584314956869158737690626, −7.32452907911919048961393137115, −7.00789836602816516380089459930, −4.57532517162768516512494890044, −3.69963751770568574543801711943, −2.65788896685577646673971452066, −0.53007661579108166128813225438, 2.58129185142838703156570745255, 3.38918379580458723295999237943, 5.28597590580298054013400735640, 6.14025135347473913600510209914, 7.78935859365538656998239851622, 8.120543445770525780130191525937, 8.703843800992040330185163005712, 10.49481150467019142227437390282, 11.47903898955260195994098009310, 12.08111097593651240780614013565

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