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Properties

Label 2-105-35.4-c3-0-10
Degree $2$
Conductor $105$
Sign $0.255 - 0.966i$
Analytic cond. $6.19520$
Root an. cond. $2.48901$
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  + (1.50 − 0.866i)2-s + (2.59 + 1.5i)3-s + (−2.49 + 4.32i)4-s + (−8.24 + 7.54i)5-s + 5.19·6-s + (14.9 − 10.9i)7-s + 22.5i·8-s + (4.5 + 7.79i)9-s + (−5.83 + 18.4i)10-s + (−16.1 + 28.0i)11-s + (−12.9 + 7.49i)12-s + 71.5i·13-s + (12.8 − 29.3i)14-s + (−32.7 + 7.24i)15-s + (−0.474 − 0.821i)16-s + (79.8 + 46.1i)17-s + ⋯
L(s)  = 1  + (0.530 − 0.306i)2-s + (0.499 + 0.288i)3-s + (−0.312 + 0.540i)4-s + (−0.737 + 0.675i)5-s + 0.353·6-s + (0.805 − 0.592i)7-s + 0.995i·8-s + (0.166 + 0.288i)9-s + (−0.184 + 0.584i)10-s + (−0.443 + 0.767i)11-s + (−0.312 + 0.180i)12-s + 1.52i·13-s + (0.245 − 0.561i)14-s + (−0.563 + 0.124i)15-s + (−0.00740 − 0.0128i)16-s + (1.13 + 0.657i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.255 - 0.966i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.255 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.255 - 0.966i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.255 - 0.966i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.55597 + 1.19800i\)
\(L(\frac12)\) \(\approx\) \(1.55597 + 1.19800i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.59 - 1.5i)T \)
5 \( 1 + (8.24 - 7.54i)T \)
7 \( 1 + (-14.9 + 10.9i)T \)
good2 \( 1 + (-1.50 + 0.866i)T + (4 - 6.92i)T^{2} \)
11 \( 1 + (16.1 - 28.0i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 71.5iT - 2.19e3T^{2} \)
17 \( 1 + (-79.8 - 46.1i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (69.3 + 120. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-104. + 60.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 4.48T + 2.43e4T^{2} \)
31 \( 1 + (-9.30 + 16.1i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-237. + 136. i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 168.T + 6.89e4T^{2} \)
43 \( 1 + 176. iT - 7.95e4T^{2} \)
47 \( 1 + (151. - 87.6i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-213. - 123. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (381. - 661. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-69.6 - 120. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (133. + 77.2i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 610.T + 3.57e5T^{2} \)
73 \( 1 + (-952. - 550. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (642. + 1.11e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 264. iT - 5.71e5T^{2} \)
89 \( 1 + (-366. - 635. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 772. iT - 9.12e5T^{2} \)
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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

−13.58706163909250510499759471844, −12.48224552946606452893104905370, −11.41140065069492537937589043215, −10.63216185333588571607393844360, −9.049255255107502942940807275331, −7.933388954501905062617351820697, −7.05912014758576913892343736339, −4.67802540169332197402874919209, −4.01486875034113811312054574159, −2.48573699500476091951399850261, 0.971955188083738465344559806640, 3.38947622920372847578662420232, 4.98439691729588961451827836870, 5.81229485852728796853371639266, 7.77991106657952725212400124546, 8.369716255648979952581784746425, 9.713699936886080154564526684537, 11.05754661756821194747336535726, 12.40833166235801617852446459501, 13.04727854247596742722234464208

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