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Properties

Label 2-1050-35.13-c1-0-1
Degree $2$
Conductor $1050$
Sign $-0.865 - 0.500i$
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  + (−0.707 + 0.707i)2-s + (−0.707 + 0.707i)3-s − 1.00i·4-s − 1.00i·6-s + (−0.189 + 2.63i)7-s + (0.707 + 0.707i)8-s − 1.00i·9-s + 1.46·11-s + (0.707 + 0.707i)12-s + (−0.189 + 0.189i)13-s + (−1.73 − 2i)14-s − 1.00·16-s + (3.53 + 3.53i)17-s + (0.707 + 0.707i)18-s + 0.535·19-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.408 + 0.408i)3-s − 0.500i·4-s − 0.408i·6-s + (−0.0716 + 0.997i)7-s + (0.250 + 0.250i)8-s − 0.333i·9-s + 0.441·11-s + (0.204 + 0.204i)12-s + (−0.0525 + 0.0525i)13-s + (−0.462 − 0.534i)14-s − 0.250·16-s + (0.857 + 0.857i)17-s + (0.166 + 0.166i)18-s + 0.122·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 - 0.500i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.865 - 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.865 - 0.500i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.865 - 0.500i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8401893708\)
\(L(\frac12)\) \(\approx\) \(0.8401893708\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 \)
7 \( 1 + (0.189 - 2.63i)T \)
good11 \( 1 - 1.46T + 11T^{2} \)
13 \( 1 + (0.189 - 0.189i)T - 13iT^{2} \)
17 \( 1 + (-3.53 - 3.53i)T + 17iT^{2} \)
19 \( 1 - 0.535T + 19T^{2} \)
23 \( 1 + (0.707 + 0.707i)T + 23iT^{2} \)
29 \( 1 + 4.26iT - 29T^{2} \)
31 \( 1 - 5.92iT - 31T^{2} \)
37 \( 1 + (4.89 - 4.89i)T - 37iT^{2} \)
41 \( 1 - 2.26iT - 41T^{2} \)
43 \( 1 + (0.189 + 0.189i)T + 43iT^{2} \)
47 \( 1 + (-8.76 - 8.76i)T + 47iT^{2} \)
53 \( 1 + (7.39 + 7.39i)T + 53iT^{2} \)
59 \( 1 + 3.19T + 59T^{2} \)
61 \( 1 - 4.46iT - 61T^{2} \)
67 \( 1 + (10.1 - 10.1i)T - 67iT^{2} \)
71 \( 1 + 4.53T + 71T^{2} \)
73 \( 1 + (5.93 - 5.93i)T - 73iT^{2} \)
79 \( 1 - 8.39iT - 79T^{2} \)
83 \( 1 + (-4.57 + 4.57i)T - 83iT^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 + (11.9 + 11.9i)T + 97iT^{2} \)
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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

−10.11318504497507193596609729940, −9.397807357879314943684427087590, −8.649269689641959531836669500388, −7.919851412078298779189853583021, −6.77517681259729663321085750827, −6.00290651019549668354062805912, −5.35582481321847059841301770486, −4.28089022788708433385095266303, −2.99894127911025933011368557703, −1.47329313818792461910620134636, 0.50517866845657042615610238048, 1.67192378496122452958921133837, 3.12246173609424849319508328089, 4.10451309398996819841073958837, 5.21715349116391373529194533429, 6.33632884325203006147325571141, 7.35008160292367124880310136353, 7.66063431999627594503622082456, 8.903704907353346469220421353752, 9.648086451407165957641879785168

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