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Properties

Label 2-51-17.13-c5-0-14
Degree $2$
Conductor $51$
Sign $-0.0688 - 0.997i$
Analytic cond. $8.17957$
Root an. cond. $2.85999$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9.80i·2-s + (6.36 + 6.36i)3-s − 64.1·4-s + (−34.6 − 34.6i)5-s + (62.3 − 62.3i)6-s + (−73.5 + 73.5i)7-s + 314. i·8-s + 81i·9-s + (−339. + 339. i)10-s + (−129. + 129. i)11-s + (−408. − 408. i)12-s − 732.·13-s + (721. + 721. i)14-s − 440. i·15-s + 1.03e3·16-s + (1.18e3 + 63.7i)17-s + ⋯
L(s)  = 1  − 1.73i·2-s + (0.408 + 0.408i)3-s − 2.00·4-s + (−0.619 − 0.619i)5-s + (0.707 − 0.707i)6-s + (−0.567 + 0.567i)7-s + 1.73i·8-s + 0.333i·9-s + (−1.07 + 1.07i)10-s + (−0.322 + 0.322i)11-s + (−0.817 − 0.817i)12-s − 1.20·13-s + (0.983 + 0.983i)14-s − 0.505i·15-s + 1.01·16-s + (0.998 + 0.0535i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0688 - 0.997i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0688 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $-0.0688 - 0.997i$
Analytic conductor: \(8.17957\)
Root analytic conductor: \(2.85999\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{51} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 51,\ (\ :5/2),\ -0.0688 - 0.997i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.177074 + 0.189722i\)
\(L(\frac12)\) \(\approx\) \(0.177074 + 0.189722i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-6.36 - 6.36i)T \)
17 \( 1 + (-1.18e3 - 63.7i)T \)
good2 \( 1 + 9.80iT - 32T^{2} \)
5 \( 1 + (34.6 + 34.6i)T + 3.12e3iT^{2} \)
7 \( 1 + (73.5 - 73.5i)T - 1.68e4iT^{2} \)
11 \( 1 + (129. - 129. i)T - 1.61e5iT^{2} \)
13 \( 1 + 732.T + 3.71e5T^{2} \)
19 \( 1 + 560. iT - 2.47e6T^{2} \)
23 \( 1 + (512. - 512. i)T - 6.43e6iT^{2} \)
29 \( 1 + (5.83e3 + 5.83e3i)T + 2.05e7iT^{2} \)
31 \( 1 + (3.61e3 + 3.61e3i)T + 2.86e7iT^{2} \)
37 \( 1 + (5.12e3 + 5.12e3i)T + 6.93e7iT^{2} \)
41 \( 1 + (1.25e4 - 1.25e4i)T - 1.15e8iT^{2} \)
43 \( 1 - 9.58e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.42e4T + 2.29e8T^{2} \)
53 \( 1 + 2.30e4iT - 4.18e8T^{2} \)
59 \( 1 + 4.55e4iT - 7.14e8T^{2} \)
61 \( 1 + (-2.49e4 + 2.49e4i)T - 8.44e8iT^{2} \)
67 \( 1 + 1.09e4T + 1.35e9T^{2} \)
71 \( 1 + (-5.11e4 - 5.11e4i)T + 1.80e9iT^{2} \)
73 \( 1 + (4.00e4 + 4.00e4i)T + 2.07e9iT^{2} \)
79 \( 1 + (6.28e3 - 6.28e3i)T - 3.07e9iT^{2} \)
83 \( 1 - 6.11e4iT - 3.93e9T^{2} \)
89 \( 1 - 6.57e4T + 5.58e9T^{2} \)
97 \( 1 + (-9.66e3 - 9.66e3i)T + 8.58e9iT^{2} \)
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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

−13.10252356417845262194093401665, −12.36584962358907856618832902643, −11.46188617648169266920825133986, −9.960039801797910149195533793846, −9.367772069834767349263870134001, −7.949130060184194514542282582235, −5.04369378354402474844733224394, −3.70862364111833554231448627578, −2.31209394080252706030510482567, −0.12111631591379361624544480016, 3.55066767629714807197971877933, 5.46598128519770292817916938167, 7.13164750742550791995402074667, 7.41584689146268294434980287660, 8.839122650950756568271244286324, 10.29543618720814659717941764770, 12.28729378511467712874026124956, 13.57763575199495732726525692259, 14.50585376044002133669040462022, 15.20222403885507760181642361076

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