Properties

Label 2-105-3.2-c2-0-5
Degree $2$
Conductor $105$
Sign $0.769 - 0.638i$
Analytic cond. $2.86104$
Root an. cond. $1.69146$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.209i·2-s + (1.91 + 2.30i)3-s + 3.95·4-s + 2.23i·5-s + (0.483 − 0.400i)6-s − 2.64·7-s − 1.66i·8-s + (−1.66 + 8.84i)9-s + 0.467·10-s − 9.19i·11-s + (7.57 + 9.13i)12-s − 7.18·13-s + 0.553i·14-s + (−5.16 + 4.28i)15-s + 15.4·16-s + 4.35i·17-s + ⋯
L(s)  = 1  − 0.104i·2-s + (0.638 + 0.769i)3-s + 0.989·4-s + 0.447i·5-s + (0.0805 − 0.0668i)6-s − 0.377·7-s − 0.208i·8-s + (−0.184 + 0.982i)9-s + 0.0467·10-s − 0.836i·11-s + (0.631 + 0.761i)12-s − 0.552·13-s + 0.0395i·14-s + (−0.344 + 0.285i)15-s + 0.967·16-s + 0.256i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.769 - 0.638i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.769 - 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.769 - 0.638i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.769 - 0.638i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.74174 + 0.628414i\)
\(L(\frac12)\) \(\approx\) \(1.74174 + 0.628414i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.91 - 2.30i)T \)
5 \( 1 - 2.23iT \)
7 \( 1 + 2.64T \)
good2 \( 1 + 0.209iT - 4T^{2} \)
11 \( 1 + 9.19iT - 121T^{2} \)
13 \( 1 + 7.18T + 169T^{2} \)
17 \( 1 - 4.35iT - 289T^{2} \)
19 \( 1 - 20.7T + 361T^{2} \)
23 \( 1 + 26.5iT - 529T^{2} \)
29 \( 1 + 27.3iT - 841T^{2} \)
31 \( 1 + 33.2T + 961T^{2} \)
37 \( 1 + 58.4T + 1.36e3T^{2} \)
41 \( 1 + 39.8iT - 1.68e3T^{2} \)
43 \( 1 - 23.7T + 1.84e3T^{2} \)
47 \( 1 - 85.1iT - 2.20e3T^{2} \)
53 \( 1 + 20.7iT - 2.80e3T^{2} \)
59 \( 1 + 72.4iT - 3.48e3T^{2} \)
61 \( 1 - 86.5T + 3.72e3T^{2} \)
67 \( 1 + 38.8T + 4.48e3T^{2} \)
71 \( 1 - 91.4iT - 5.04e3T^{2} \)
73 \( 1 - 11.4T + 5.32e3T^{2} \)
79 \( 1 + 41.5T + 6.24e3T^{2} \)
83 \( 1 - 154. iT - 6.88e3T^{2} \)
89 \( 1 - 135. iT - 7.92e3T^{2} \)
97 \( 1 + 48.0T + 9.40e3T^{2} \)
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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

−13.92607287013474258702367635179, −12.51862287021195020358678906749, −11.27372354401119795132810488866, −10.49961800102576455625521656089, −9.519847824719378994718762345779, −8.149411537543650747042626240373, −6.98463865260224924316445683413, −5.58074138315071512305782910656, −3.64271857080509404155600147589, −2.52931214945065250400293905416, 1.74090230072697758871809786724, 3.27058038914000002417440862236, 5.48457651235620442475702162187, 7.01944471594018235012715557654, 7.52208007093287816892494221489, 8.995177477406343482430409376271, 10.06601188859482907913875060179, 11.68655379121452042042093914787, 12.33211077196820859136670555256, 13.32080066697058300269944897497

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