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Properties

Label 2-253-11.2-c2-0-13
Degree $2$
Conductor $253$
Sign $0.508 - 0.860i$
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  + (−2.81 − 0.913i)2-s + (−3.21 + 2.33i)3-s + (3.83 + 2.78i)4-s + (1.38 + 4.25i)5-s + (11.1 − 3.62i)6-s + (3.16 − 4.35i)7-s + (−1.29 − 1.78i)8-s + (2.09 − 6.43i)9-s − 13.2i·10-s + (8.08 − 7.45i)11-s − 18.8·12-s + (−16.0 − 5.21i)13-s + (−12.8 + 9.36i)14-s + (−14.3 − 10.4i)15-s + (−3.85 − 11.8i)16-s + (15.0 − 4.89i)17-s + ⋯
L(s)  = 1  + (−1.40 − 0.456i)2-s + (−1.07 + 0.777i)3-s + (0.959 + 0.697i)4-s + (0.276 + 0.850i)5-s + (1.86 − 0.604i)6-s + (0.452 − 0.622i)7-s + (−0.162 − 0.223i)8-s + (0.232 − 0.714i)9-s − 1.32i·10-s + (0.735 − 0.677i)11-s − 1.57·12-s + (−1.23 − 0.400i)13-s + (−0.920 + 0.668i)14-s + (−0.957 − 0.695i)15-s + (−0.240 − 0.740i)16-s + (0.886 − 0.287i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 - 0.860i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.508 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(253\)    =    \(11 \cdot 23\)
Sign: $0.508 - 0.860i$
Analytic conductor: \(6.89375\)
Root analytic conductor: \(2.62559\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{253} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 253,\ (\ :1),\ 0.508 - 0.860i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.467343 + 0.266636i\)
\(L(\frac12)\) \(\approx\) \(0.467343 + 0.266636i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-8.08 + 7.45i)T \)
23 \( 1 - 4.79T \)
good2 \( 1 + (2.81 + 0.913i)T + (3.23 + 2.35i)T^{2} \)
3 \( 1 + (3.21 - 2.33i)T + (2.78 - 8.55i)T^{2} \)
5 \( 1 + (-1.38 - 4.25i)T + (-20.2 + 14.6i)T^{2} \)
7 \( 1 + (-3.16 + 4.35i)T + (-15.1 - 46.6i)T^{2} \)
13 \( 1 + (16.0 + 5.21i)T + (136. + 99.3i)T^{2} \)
17 \( 1 + (-15.0 + 4.89i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (-18.5 - 25.5i)T + (-111. + 343. i)T^{2} \)
29 \( 1 + (-17.8 + 24.6i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (13.7 - 42.2i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (4.03 + 2.93i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (-41.3 - 56.9i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 - 54.5iT - 1.84e3T^{2} \)
47 \( 1 + (9.04 - 6.57i)T + (682. - 2.10e3i)T^{2} \)
53 \( 1 + (-6.48 + 19.9i)T + (-2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (59.2 + 43.0i)T + (1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (-30.6 + 9.96i)T + (3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 + 66.0T + 4.48e3T^{2} \)
71 \( 1 + (-22.6 - 69.6i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (77.8 - 107. i)T + (-1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (15.1 + 4.90i)T + (5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (-6.22 + 2.02i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 - 52.0T + 7.92e3T^{2} \)
97 \( 1 + (4.28 - 13.1i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−11.46021629782829226934493668243, −10.81434696319182283295773789105, −10.01926377377896841967085093410, −9.695597444480865373741886799668, −8.123640782925085779124594521248, −7.26279753555382033255388512412, −5.94755902397906685090613647496, −4.75862780817747009544192560400, −3.05403378855627756868303237545, −1.06795162476902262614131638490, 0.67061805038389503287564807339, 1.81434037093056131838940782472, 4.82069852907004932613693612158, 5.76087191331044055662867717081, 7.03409717000178973070660370028, 7.50207359532486480994057459385, 8.994814772348613689354405105041, 9.325347569311016096878203730275, 10.57007814470436938282021650397, 11.80576841425591081172100608739

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