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Properties

Label 2-930-5.4-c3-0-83
Degree $2$
Conductor $930$
Sign $-0.557 - 0.830i$
Analytic cond. $54.8717$
Root an. cond. $7.40754$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s + 3i·3-s − 4·4-s + (−6.23 − 9.28i)5-s + 6·6-s − 2.24i·7-s + 8i·8-s − 9·9-s + (−18.5 + 12.4i)10-s + 10.8·11-s − 12i·12-s + 9.95i·13-s − 4.49·14-s + (27.8 − 18.7i)15-s + 16·16-s − 47.5i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (−0.557 − 0.830i)5-s + 0.408·6-s − 0.121i·7-s + 0.353i·8-s − 0.333·9-s + (−0.586 + 0.394i)10-s + 0.298·11-s − 0.288i·12-s + 0.212i·13-s − 0.0859·14-s + (0.479 − 0.321i)15-s + 0.250·16-s − 0.678i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.557 - 0.830i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.557 - 0.830i$
Analytic conductor: \(54.8717\)
Root analytic conductor: \(7.40754\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :3/2),\ -0.557 - 0.830i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1082960206\)
\(L(\frac12)\) \(\approx\) \(0.1082960206\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2iT \)
3 \( 1 - 3iT \)
5 \( 1 + (6.23 + 9.28i)T \)
31 \( 1 + 31T \)
good7 \( 1 + 2.24iT - 343T^{2} \)
11 \( 1 - 10.8T + 1.33e3T^{2} \)
13 \( 1 - 9.95iT - 2.19e3T^{2} \)
17 \( 1 + 47.5iT - 4.91e3T^{2} \)
19 \( 1 - 25.1T + 6.85e3T^{2} \)
23 \( 1 + 205. iT - 1.21e4T^{2} \)
29 \( 1 - 98.6T + 2.43e4T^{2} \)
37 \( 1 - 130. iT - 5.06e4T^{2} \)
41 \( 1 + 482.T + 6.89e4T^{2} \)
43 \( 1 + 274. iT - 7.95e4T^{2} \)
47 \( 1 - 155. iT - 1.03e5T^{2} \)
53 \( 1 - 205. iT - 1.48e5T^{2} \)
59 \( 1 + 449.T + 2.05e5T^{2} \)
61 \( 1 + 402.T + 2.26e5T^{2} \)
67 \( 1 - 709. iT - 3.00e5T^{2} \)
71 \( 1 - 92.4T + 3.57e5T^{2} \)
73 \( 1 - 207. iT - 3.89e5T^{2} \)
79 \( 1 + 391.T + 4.93e5T^{2} \)
83 \( 1 + 936. iT - 5.71e5T^{2} \)
89 \( 1 - 500.T + 7.04e5T^{2} \)
97 \( 1 + 418. iT - 9.12e5T^{2} \)
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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

−9.041064387257898930070056526251, −8.688883548925148436666812615995, −7.67331686646989320443291604533, −6.50250186223627623737742100977, −5.18802499108099826962039611883, −4.56327503856829663347586290287, −3.75168157423045658987191113913, −2.64320537918927664051052177466, −1.15713060591730730909109165720, −0.03086997531966733660101505266, 1.55765323418944905859859934425, 3.06899139780350368836460631586, 3.91072254985646999762511748377, 5.23515338349259156726914858004, 6.15211192776355512554172335536, 6.88931580637545907578050722385, 7.63608072606010667816121179820, 8.240618465634143830833425257571, 9.246137150507021268706601729904, 10.17459083803163555574219423714

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