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Properties

Label 2-51-51.20-c1-0-1
Degree $2$
Conductor $51$
Sign $0.969 + 0.244i$
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  + (−0.295 − 0.713i)2-s + (−0.521 + 1.65i)3-s + (0.992 − 0.992i)4-s + (2.16 − 1.44i)5-s + (1.33 − 0.116i)6-s + (−2.14 + 3.21i)7-s + (−2.42 − 1.00i)8-s + (−2.45 − 1.72i)9-s + (−1.66 − 1.11i)10-s + (−2.20 + 0.438i)11-s + (1.12 + 2.15i)12-s + (−1.56 − 1.56i)13-s + (2.92 + 0.582i)14-s + (1.25 + 4.32i)15-s − 0.777i·16-s + (1.07 + 3.97i)17-s + ⋯
L(s)  = 1  + (−0.208 − 0.504i)2-s + (−0.300 + 0.953i)3-s + (0.496 − 0.496i)4-s + (0.966 − 0.645i)5-s + (0.543 − 0.0475i)6-s + (−0.811 + 1.21i)7-s + (−0.858 − 0.355i)8-s + (−0.818 − 0.573i)9-s + (−0.527 − 0.352i)10-s + (−0.665 + 0.132i)11-s + (0.323 + 0.622i)12-s + (−0.435 − 0.435i)13-s + (0.782 + 0.155i)14-s + (0.324 + 1.11i)15-s − 0.194i·16-s + (0.261 + 0.965i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.244i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 + 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $0.969 + 0.244i$
Analytic conductor: \(0.407237\)
Root analytic conductor: \(0.638151\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{51} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 51,\ (\ :1/2),\ 0.969 + 0.244i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.775592 - 0.0963068i\)
\(L(\frac12)\) \(\approx\) \(0.775592 - 0.0963068i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.521 - 1.65i)T \)
17 \( 1 + (-1.07 - 3.97i)T \)
good2 \( 1 + (0.295 + 0.713i)T + (-1.41 + 1.41i)T^{2} \)
5 \( 1 + (-2.16 + 1.44i)T + (1.91 - 4.61i)T^{2} \)
7 \( 1 + (2.14 - 3.21i)T + (-2.67 - 6.46i)T^{2} \)
11 \( 1 + (2.20 - 0.438i)T + (10.1 - 4.20i)T^{2} \)
13 \( 1 + (1.56 + 1.56i)T + 13iT^{2} \)
19 \( 1 + (-2.41 + 1.00i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (0.751 + 3.77i)T + (-21.2 + 8.80i)T^{2} \)
29 \( 1 + (-2.34 - 3.50i)T + (-11.0 + 26.7i)T^{2} \)
31 \( 1 + (-0.352 + 1.77i)T + (-28.6 - 11.8i)T^{2} \)
37 \( 1 + (-3.53 - 0.702i)T + (34.1 + 14.1i)T^{2} \)
41 \( 1 + (-1.47 - 0.985i)T + (15.6 + 37.8i)T^{2} \)
43 \( 1 + (-2.04 - 0.846i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + (-6.39 + 6.39i)T - 47iT^{2} \)
53 \( 1 + (0.551 + 1.33i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (6.77 + 2.80i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-3.25 - 2.17i)T + (23.3 + 56.3i)T^{2} \)
67 \( 1 - 11.5iT - 67T^{2} \)
71 \( 1 + (1.30 - 6.55i)T + (-65.5 - 27.1i)T^{2} \)
73 \( 1 + (8.09 + 12.1i)T + (-27.9 + 67.4i)T^{2} \)
79 \( 1 + (0.837 + 4.21i)T + (-72.9 + 30.2i)T^{2} \)
83 \( 1 + (-3.78 + 1.56i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (6.55 + 6.55i)T + 89iT^{2} \)
97 \( 1 + (11.3 - 7.55i)T + (37.1 - 89.6i)T^{2} \)
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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

−15.58003636687918291369654811720, −14.69964215960565255482520605040, −12.88115485514158291009341331703, −11.99068201616171807804523640000, −10.49528990820316048398249885170, −9.760219535066321340631230597394, −8.874412096848746304640755841343, −6.09782086240658436731314778797, −5.34644687023263760649569822370, −2.70538895259498361311649468465, 2.78283068327502491618018568805, 5.91292329319461636993519761043, 6.96241765837349996622031909866, 7.67229779041808962622935506438, 9.671159079514065833770701027753, 10.95458732040592257448237193063, 12.25330692510184253982684603133, 13.50949774548979543587620250761, 14.11909470801263957895420598584, 15.92285160468734584349056550069

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