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Properties

Label 2-1050-105.104-c1-0-27
Degree $2$
Conductor $1050$
Sign $0.850 - 0.525i$
Analytic cond. $8.38429$
Root an. cond. $2.89556$
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  + 2-s + (1.57 + 0.721i)3-s + 4-s + (1.57 + 0.721i)6-s + (2.29 − 1.31i)7-s + 8-s + (1.95 + 2.27i)9-s + 3.91i·11-s + (1.57 + 0.721i)12-s − 4.99·13-s + (2.29 − 1.31i)14-s + 16-s + 3.54i·17-s + (1.95 + 2.27i)18-s − 3.14i·19-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.909 + 0.416i)3-s + 0.5·4-s + (0.642 + 0.294i)6-s + (0.867 − 0.496i)7-s + 0.353·8-s + (0.652 + 0.757i)9-s + 1.18i·11-s + (0.454 + 0.208i)12-s − 1.38·13-s + (0.613 − 0.351i)14-s + 0.250·16-s + 0.860i·17-s + (0.461 + 0.535i)18-s − 0.722i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (1049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.850 - 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.599786633\)
\(L(\frac12)\) \(\approx\) \(3.599786633\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-1.57 - 0.721i)T \)
5 \( 1 \)
7 \( 1 + (-2.29 + 1.31i)T \)
good11 \( 1 - 3.91iT - 11T^{2} \)
13 \( 1 + 4.99T + 13T^{2} \)
17 \( 1 - 3.54iT - 17T^{2} \)
19 \( 1 + 3.14iT - 19T^{2} \)
23 \( 1 - 7.54T + 23T^{2} \)
29 \( 1 + 7.54iT - 29T^{2} \)
31 \( 1 - 4.19iT - 31T^{2} \)
37 \( 1 + 10.4iT - 37T^{2} \)
41 \( 1 + 9.32T + 41T^{2} \)
43 \( 1 - 2.91iT - 43T^{2} \)
47 \( 1 + 8.00iT - 47T^{2} \)
53 \( 1 + 0.288T + 53T^{2} \)
59 \( 1 + 5.89T + 59T^{2} \)
61 \( 1 - 2.48iT - 61T^{2} \)
67 \( 1 - 0.545iT - 67T^{2} \)
71 \( 1 - 5.37iT - 71T^{2} \)
73 \( 1 - 4.85T + 73T^{2} \)
79 \( 1 + 0.742T + 79T^{2} \)
83 \( 1 + 4.45iT - 83T^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 + 0.524T + 97T^{2} \)
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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

−10.06633061171544159899091028623, −9.214503379631296788646429053900, −8.229113580418801753059488555446, −7.34656945460549348019265033976, −6.95252730525519930050833791991, −5.21698851324653698712306666191, −4.68922152510749766653841881123, −3.93068172163531910796164885847, −2.65826475882876974152882287679, −1.79706509620442974308830948049, 1.39945618027725773384606748354, 2.67125908763428994749090689218, 3.29063580519127937575970038627, 4.69497284664499641867530557594, 5.33061330869725299085076577720, 6.54395833369080102687373336356, 7.36167418722789978101541170061, 8.119963738231020210090555412707, 8.873651215762974944864539359371, 9.711542937415637899145048053443

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