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Properties

Label 2-276-4.3-c2-0-34
Degree $2$
Conductor $276$
Sign $0.807 + 0.589i$
Analytic cond. $7.52045$
Root an. cond. $2.74234$
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  + (1.90 + 0.620i)2-s − 1.73i·3-s + (3.23 + 2.35i)4-s + 0.944·5-s + (1.07 − 3.29i)6-s − 12.0i·7-s + (4.67 + 6.48i)8-s − 2.99·9-s + (1.79 + 0.585i)10-s − 12.0i·11-s + (4.08 − 5.59i)12-s + 1.77·13-s + (7.44 − 22.8i)14-s − 1.63i·15-s + (4.86 + 15.2i)16-s + 22.5·17-s + ⋯
L(s)  = 1  + (0.950 + 0.310i)2-s − 0.577i·3-s + (0.807 + 0.589i)4-s + 0.188·5-s + (0.179 − 0.548i)6-s − 1.71i·7-s + (0.584 + 0.811i)8-s − 0.333·9-s + (0.179 + 0.0585i)10-s − 1.09i·11-s + (0.340 − 0.466i)12-s + 0.136·13-s + (0.531 − 1.63i)14-s − 0.109i·15-s + (0.304 + 0.952i)16-s + 1.32·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 + 0.589i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.807 + 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $0.807 + 0.589i$
Analytic conductor: \(7.52045\)
Root analytic conductor: \(2.74234\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{276} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 276,\ (\ :1),\ 0.807 + 0.589i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.80346 - 0.914689i\)
\(L(\frac12)\) \(\approx\) \(2.80346 - 0.914689i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.90 - 0.620i)T \)
3 \( 1 + 1.73iT \)
23 \( 1 + 4.79iT \)
good5 \( 1 - 0.944T + 25T^{2} \)
7 \( 1 + 12.0iT - 49T^{2} \)
11 \( 1 + 12.0iT - 121T^{2} \)
13 \( 1 - 1.77T + 169T^{2} \)
17 \( 1 - 22.5T + 289T^{2} \)
19 \( 1 - 28.0iT - 361T^{2} \)
29 \( 1 - 0.588T + 841T^{2} \)
31 \( 1 + 4.11iT - 961T^{2} \)
37 \( 1 - 72.4T + 1.36e3T^{2} \)
41 \( 1 + 31.9T + 1.68e3T^{2} \)
43 \( 1 - 18.0iT - 1.84e3T^{2} \)
47 \( 1 - 19.7iT - 2.20e3T^{2} \)
53 \( 1 + 62.3T + 2.80e3T^{2} \)
59 \( 1 - 24.6iT - 3.48e3T^{2} \)
61 \( 1 + 78.1T + 3.72e3T^{2} \)
67 \( 1 - 110. iT - 4.48e3T^{2} \)
71 \( 1 + 16.9iT - 5.04e3T^{2} \)
73 \( 1 - 80.0T + 5.32e3T^{2} \)
79 \( 1 - 134. iT - 6.24e3T^{2} \)
83 \( 1 + 120. iT - 6.88e3T^{2} \)
89 \( 1 - 76.9T + 7.92e3T^{2} \)
97 \( 1 + 79.5T + 9.40e3T^{2} \)
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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

−11.71117817847865073955546634862, −10.85247067719664515388406533166, −9.930229895474073452927685532028, −8.016420831293119438924922318680, −7.69133548793522404690736862153, −6.42956036681124046864351347648, −5.68341857184919063050209355641, −4.14411870905784989276950054218, −3.23509549146195336368712770451, −1.26902528273015873429202322704, 2.08545084601028785636764305919, 3.15823901812791034407683273898, 4.67596615939723741175783506996, 5.45092409946892449449627551360, 6.37500749374412311694260037555, 7.80935180726104766288780215913, 9.299834032693721806361166354657, 9.803000519863402305272704553066, 11.06733014046747490168969841215, 11.92600300199722932574247476215

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