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Properties

Label 2-1050-105.104-c1-0-37
Degree $2$
Conductor $1050$
Sign $0.961 + 0.273i$
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 + (−0.403 + 1.68i)3-s + 4-s + (−0.403 + 1.68i)6-s + (1.28 − 2.31i)7-s + 8-s + (−2.67 − 1.35i)9-s − 5.34i·11-s + (−0.403 + 1.68i)12-s + 3.95·13-s + (1.28 − 2.31i)14-s + 16-s − 7.32i·17-s + (−2.67 − 1.35i)18-s + 0.807i·19-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.232 + 0.972i)3-s + 0.5·4-s + (−0.164 + 0.687i)6-s + (0.484 − 0.874i)7-s + 0.353·8-s + (−0.891 − 0.453i)9-s − 1.61i·11-s + (−0.116 + 0.486i)12-s + 1.09·13-s + (0.342 − 0.618i)14-s + 0.250·16-s − 1.77i·17-s + (−0.630 − 0.320i)18-s + 0.185i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.273i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 + 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.961 + 0.273i$
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: \((2,\ 1050,\ (\ :1/2),\ 0.961 + 0.273i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.384705494\)
\(L(\frac12)\) \(\approx\) \(2.384705494\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.403 - 1.68i)T \)
5 \( 1 \)
7 \( 1 + (-1.28 + 2.31i)T \)
good11 \( 1 + 5.34iT - 11T^{2} \)
13 \( 1 - 3.95T + 13T^{2} \)
17 \( 1 + 7.32iT - 17T^{2} \)
19 \( 1 - 0.807iT - 19T^{2} \)
23 \( 1 - 0.281T + 23T^{2} \)
29 \( 1 + 0.281iT - 29T^{2} \)
31 \( 1 - 9.07iT - 31T^{2} \)
37 \( 1 - 6.06iT - 37T^{2} \)
41 \( 1 + 6.15T + 41T^{2} \)
43 \( 1 + 6.34iT - 43T^{2} \)
47 \( 1 - 5.78iT - 47T^{2} \)
53 \( 1 - 10.9T + 53T^{2} \)
59 \( 1 + 4.90T + 59T^{2} \)
61 \( 1 - 13.2iT - 61T^{2} \)
67 \( 1 + 6.71iT - 67T^{2} \)
71 \( 1 - 3.36iT - 71T^{2} \)
73 \( 1 + 4.98T + 73T^{2} \)
79 \( 1 - 3.26T + 79T^{2} \)
83 \( 1 + 1.53iT - 83T^{2} \)
89 \( 1 + 4.31T + 89T^{2} \)
97 \( 1 - 15.0T + 97T^{2} \)
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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

−10.23552516392806122412670205137, −8.957958994608959958977009909934, −8.399341450329244878552826973414, −7.20373032264968398518071442328, −6.25147333644949419143853997914, −5.38715323805954909981284549499, −4.66269203589370431526050043848, −3.62416730895101115260684975051, −3.04613986804789430790926711197, −0.946101667346293991653068634513, 1.65655707816319752522146437262, 2.27481677897418109373972890787, 3.77389318149014415181553043308, 4.84307709910350991316416719148, 5.83248480833648928288677014921, 6.34938972104046915492609020797, 7.36784527475559053009983491995, 8.127418690144326121380880281851, 8.892417355233196657452109945989, 10.12706388447924291283209314822

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