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Properties

Label 2-1530-255.47-c1-0-6
Degree $2$
Conductor $1530$
Sign $0.893 - 0.449i$
Analytic cond. $12.2171$
Root an. cond. $3.49529$
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 + 1.00i·4-s + (−1.98 − 1.02i)5-s + 1.10·7-s + (0.707 − 0.707i)8-s + (0.675 + 2.13i)10-s + (−2.77 − 2.77i)11-s + (−2.44 + 2.44i)13-s + (−0.778 − 0.778i)14-s − 1.00·16-s + (−2.61 + 3.18i)17-s − 0.349·19-s + (1.02 − 1.98i)20-s + 3.92i·22-s + 7.52·23-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−0.887 − 0.460i)5-s + 0.415·7-s + (0.250 − 0.250i)8-s + (0.213 + 0.674i)10-s + (−0.836 − 0.836i)11-s + (−0.679 + 0.679i)13-s + (−0.207 − 0.207i)14-s − 0.250·16-s + (−0.633 + 0.773i)17-s − 0.0801·19-s + (0.230 − 0.443i)20-s + 0.836i·22-s + 1.56·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.449i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 - 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1530\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $0.893 - 0.449i$
Analytic conductor: \(12.2171\)
Root analytic conductor: \(3.49529\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1530} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1530,\ (\ :1/2),\ 0.893 - 0.449i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7473606312\)
\(L(\frac12)\) \(\approx\) \(0.7473606312\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (1.98 + 1.02i)T \)
17 \( 1 + (2.61 - 3.18i)T \)
good7 \( 1 - 1.10T + 7T^{2} \)
11 \( 1 + (2.77 + 2.77i)T + 11iT^{2} \)
13 \( 1 + (2.44 - 2.44i)T - 13iT^{2} \)
19 \( 1 + 0.349T + 19T^{2} \)
23 \( 1 - 7.52T + 23T^{2} \)
29 \( 1 + (3.76 + 3.76i)T + 29iT^{2} \)
31 \( 1 + (-6.62 - 6.62i)T + 31iT^{2} \)
37 \( 1 + 8.31iT - 37T^{2} \)
41 \( 1 + (-3.79 - 3.79i)T + 41iT^{2} \)
43 \( 1 + (-0.826 - 0.826i)T + 43iT^{2} \)
47 \( 1 + (4.79 - 4.79i)T - 47iT^{2} \)
53 \( 1 + (-1.01 + 1.01i)T - 53iT^{2} \)
59 \( 1 - 8.82iT - 59T^{2} \)
61 \( 1 + (7.17 - 7.17i)T - 61iT^{2} \)
67 \( 1 + (-3.18 - 3.18i)T + 67iT^{2} \)
71 \( 1 + (-2.14 + 2.14i)T - 71iT^{2} \)
73 \( 1 - 13.2T + 73T^{2} \)
79 \( 1 + (-9.26 - 9.26i)T + 79iT^{2} \)
83 \( 1 + (-6.52 + 6.52i)T - 83iT^{2} \)
89 \( 1 - 5.03T + 89T^{2} \)
97 \( 1 - 11.4iT - 97T^{2} \)
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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

−9.320730164594816484689719976344, −8.765633184411405745237727951986, −8.026759172454261410512593634796, −7.43938910189947720292532723508, −6.41689491305705725556363481223, −5.11599702356085550871040185942, −4.44785944812705358990264895015, −3.41491195795464900775476744225, −2.37242011981007404872190481581, −0.969406349137267832615177644036, 0.43959324827177784699652969176, 2.25070665677415386992489913522, 3.23868149129133781931325339004, 4.78450126209169703186369153118, 4.98269965094373571677040641720, 6.44684682635552900938060104492, 7.20261659501745935795773834528, 7.73789182532965721388283857407, 8.368520195847885153508138715669, 9.400358523759032838092164703044

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