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Properties

Label 2-253-11.10-c2-0-9
Degree $2$
Conductor $253$
Sign $-0.340 - 0.940i$
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  + 0.431i·2-s − 1.07·3-s + 3.81·4-s − 3.78·5-s − 0.463i·6-s + 1.98i·7-s + 3.37i·8-s − 7.84·9-s − 1.63i·10-s + (3.74 + 10.3i)11-s − 4.09·12-s + 10.8i·13-s − 0.856·14-s + 4.06·15-s + 13.8·16-s + 13.6i·17-s + ⋯
L(s)  = 1  + 0.215i·2-s − 0.358·3-s + 0.953·4-s − 0.756·5-s − 0.0772i·6-s + 0.283i·7-s + 0.421i·8-s − 0.871·9-s − 0.163i·10-s + (0.340 + 0.940i)11-s − 0.341·12-s + 0.835i·13-s − 0.0611·14-s + 0.270·15-s + 0.862·16-s + 0.803i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.340 - 0.940i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(253\)    =    \(11 \cdot 23\)
Sign: $-0.340 - 0.940i$
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: \((2,\ 253,\ (\ :1),\ -0.340 - 0.940i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.666251 + 0.949385i\)
\(L(\frac12)\) \(\approx\) \(0.666251 + 0.949385i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-3.74 - 10.3i)T \)
23 \( 1 + 4.79T \)
good2 \( 1 - 0.431iT - 4T^{2} \)
3 \( 1 + 1.07T + 9T^{2} \)
5 \( 1 + 3.78T + 25T^{2} \)
7 \( 1 - 1.98iT - 49T^{2} \)
13 \( 1 - 10.8iT - 169T^{2} \)
17 \( 1 - 13.6iT - 289T^{2} \)
19 \( 1 + 5.19iT - 361T^{2} \)
29 \( 1 - 55.1iT - 841T^{2} \)
31 \( 1 + 0.0141T + 961T^{2} \)
37 \( 1 + 3.33T + 1.36e3T^{2} \)
41 \( 1 + 12.2iT - 1.68e3T^{2} \)
43 \( 1 - 0.275iT - 1.84e3T^{2} \)
47 \( 1 + 65.1T + 2.20e3T^{2} \)
53 \( 1 - 6.34T + 2.80e3T^{2} \)
59 \( 1 - 2.24T + 3.48e3T^{2} \)
61 \( 1 + 81.8iT - 3.72e3T^{2} \)
67 \( 1 - 36.3T + 4.48e3T^{2} \)
71 \( 1 - 54.9T + 5.04e3T^{2} \)
73 \( 1 + 70.4iT - 5.32e3T^{2} \)
79 \( 1 + 105. iT - 6.24e3T^{2} \)
83 \( 1 - 39.4iT - 6.88e3T^{2} \)
89 \( 1 - 73.9T + 7.92e3T^{2} \)
97 \( 1 - 47.9T + 9.40e3T^{2} \)
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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

−11.93879173483847517391524821647, −11.33913218256131308583492960709, −10.46853006107978624575186921465, −9.084480211818938385398500355323, −8.060885144961316918725155425198, −7.03188010513066811158141748947, −6.22840229144556150315435301613, −4.98745637307853400873892210617, −3.50596762190467335461259854644, −1.94492227284247683665236170164, 0.60393639772095317230579168136, 2.72963497503795060153477616871, 3.81380563216228147197269599477, 5.54075036021000908090973052271, 6.40101980880933382993347093661, 7.63619474098815650590354990283, 8.353241740430430316750408835551, 9.851087464131271696175332822004, 10.88218563008112405359940047994, 11.59152808767632855185407153807

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