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Properties

Label 2-51-17.8-c1-0-1
Degree $2$
Conductor $51$
Sign $0.880 + 0.473i$
Analytic cond. $0.407237$
Root an. cond. $0.638151$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.541 − 0.541i)2-s + (0.382 − 0.923i)3-s + 1.41i·4-s + (−1.92 − 0.796i)5-s + (−0.292 − 0.707i)6-s + (−1.14 + 0.472i)7-s + (1.84 + 1.84i)8-s + (−0.707 − 0.707i)9-s + (−1.47 + 0.609i)10-s + (−0.572 − 1.38i)11-s + (1.30 + 0.541i)12-s + 4.10i·13-s + (−0.361 + 0.873i)14-s + (−1.47 + 1.47i)15-s − 0.828·16-s + (−2.35 − 3.38i)17-s + ⋯
L(s)  = 1  + (0.382 − 0.382i)2-s + (0.220 − 0.533i)3-s + 0.707i·4-s + (−0.860 − 0.356i)5-s + (−0.119 − 0.288i)6-s + (−0.431 + 0.178i)7-s + (0.653 + 0.653i)8-s + (−0.235 − 0.235i)9-s + (−0.465 + 0.192i)10-s + (−0.172 − 0.416i)11-s + (0.377 + 0.156i)12-s + 1.13i·13-s + (−0.0966 + 0.233i)14-s + (−0.380 + 0.380i)15-s − 0.207·16-s + (−0.571 − 0.820i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $0.880 + 0.473i$
Analytic conductor: \(0.407237\)
Root analytic conductor: \(0.638151\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{51} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 51,\ (\ :1/2),\ 0.880 + 0.473i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.903781 - 0.227641i\)
\(L(\frac12)\) \(\approx\) \(0.903781 - 0.227641i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.382 + 0.923i)T \)
17 \( 1 + (2.35 + 3.38i)T \)
good2 \( 1 + (-0.541 + 0.541i)T - 2iT^{2} \)
5 \( 1 + (1.92 + 0.796i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (1.14 - 0.472i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (0.572 + 1.38i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 - 4.10iT - 13T^{2} \)
19 \( 1 + (-4.81 + 4.81i)T - 19iT^{2} \)
23 \( 1 + (-1.26 - 3.05i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-5.76 - 2.38i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-3.93 + 9.48i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (1.50 - 3.63i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (11.2 - 4.64i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (-2.43 - 2.43i)T + 43iT^{2} \)
47 \( 1 - 1.56iT - 47T^{2} \)
53 \( 1 + (2.80 - 2.80i)T - 53iT^{2} \)
59 \( 1 + (-5.70 - 5.70i)T + 59iT^{2} \)
61 \( 1 + (-3.02 + 1.25i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + 2.11T + 67T^{2} \)
71 \( 1 + (-0.0867 + 0.209i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-0.340 - 0.140i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-1.03 - 2.50i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (-10.1 + 10.1i)T - 83iT^{2} \)
89 \( 1 + 13.6iT - 89T^{2} \)
97 \( 1 + (2.48 + 1.02i)T + (68.5 + 68.5i)T^{2} \)
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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

−15.61202779644471705057663774282, −13.83675271093195614987014742184, −13.22035116391244129992141943690, −11.81796549181017723970690812402, −11.53971168732508270882067628312, −9.243282969020011855553982247823, −8.069346403184552835232449567793, −6.89791285684165008528077499080, −4.59129945222903174602847969249, −2.97869693578409368096722298073, 3.62802630143252019183747177621, 5.19623792066005013522148338510, 6.78054285265335684022407744268, 8.202237573733595773700448773034, 10.00329230337841390804242370110, 10.66019609932964105901229150983, 12.28366385459041717080399545320, 13.65713009318916519274284820206, 14.76196386409142778375349672065, 15.52442408603194403903154331822

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