Properties

Degree $2$
Conductor $1050$
Sign $-0.850 + 0.525i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s + (−1.06 − 1.36i)3-s − 4-s + (−1.36 + 1.06i)6-s + (2.62 + 0.294i)7-s + i·8-s + (−0.716 + 2.91i)9-s + 1.43i·11-s + (1.06 + 1.36i)12-s − 4.73i·13-s + (0.294 − 2.62i)14-s + 16-s + 2.59·17-s + (2.91 + 0.716i)18-s − 2.72i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.616 − 0.786i)3-s − 0.5·4-s + (−0.556 + 0.436i)6-s + (0.993 + 0.111i)7-s + 0.353i·8-s + (−0.238 + 0.971i)9-s + 0.431i·11-s + (0.308 + 0.393i)12-s − 1.31i·13-s + (0.0786 − 0.702i)14-s + 0.250·16-s + 0.630·17-s + (0.686 + 0.168i)18-s − 0.625i·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: $-0.850 + 0.525i$
Motivic weight: \(1\)
Character: $\chi_{1050} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.850 + 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.197179898\)
\(L(\frac12)\) \(\approx\) \(1.197179898\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (1.06 + 1.36i)T \)
5 \( 1 \)
7 \( 1 + (-2.62 - 0.294i)T \)
good11 \( 1 - 1.43iT - 11T^{2} \)
13 \( 1 + 4.73iT - 13T^{2} \)
17 \( 1 - 2.59T + 17T^{2} \)
19 \( 1 + 2.72iT - 19T^{2} \)
23 \( 1 + 2.82iT - 23T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 + 5.91iT - 31T^{2} \)
37 \( 1 - 2.39T + 37T^{2} \)
41 \( 1 + 11.1T + 41T^{2} \)
43 \( 1 + 0.432T + 43T^{2} \)
47 \( 1 - 10.3T + 47T^{2} \)
53 \( 1 + 5.69iT - 53T^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 + 1.05iT - 61T^{2} \)
67 \( 1 + 9.82T + 67T^{2} \)
71 \( 1 - 13.2iT - 71T^{2} \)
73 \( 1 + 7.58iT - 73T^{2} \)
79 \( 1 - 16.5T + 79T^{2} \)
83 \( 1 + 12.9T + 83T^{2} \)
89 \( 1 + 3.90T + 89T^{2} \)
97 \( 1 + 14.0iT - 97T^{2} \)
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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

−9.845770861484367106862379949468, −8.586972301811642052135858007300, −7.939284493433183976975446177813, −7.23411625434205299273878760606, −5.96207537463017239872487176178, −5.23179711281064534593059300231, −4.40556433566886486484527180350, −2.88436279169858277173064115710, −1.86885243236405861684298872184, −0.64688343361717860049380479484, 1.41374266327413573039296007045, 3.43265540813750440213394543886, 4.36760561986899344439775804188, 5.11825940542412547908100144445, 5.89134232057265525008172428511, 6.80961414585876860261903693050, 7.72130142409180694867605140403, 8.707917004420322322904494173300, 9.271216747618421753460424183853, 10.29136606248714454784362885674

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