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Properties

Label 2-105-7.2-c3-0-5
Degree $2$
Conductor $105$
Sign $0.306 - 0.951i$
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  + (−2.55 + 4.42i)2-s + (−1.5 − 2.59i)3-s + (−9.03 − 15.6i)4-s + (−2.5 + 4.33i)5-s + 15.3·6-s + (−2.74 − 18.3i)7-s + 51.4·8-s + (−4.5 + 7.79i)9-s + (−12.7 − 22.1i)10-s + (20.3 + 35.2i)11-s + (−27.1 + 46.9i)12-s + 70.9·13-s + (87.9 + 34.6i)14-s + 15.0·15-s + (−58.9 + 102. i)16-s + (−11.3 − 19.6i)17-s + ⋯
L(s)  = 1  + (−0.902 + 1.56i)2-s + (−0.288 − 0.499i)3-s + (−1.12 − 1.95i)4-s + (−0.223 + 0.387i)5-s + 1.04·6-s + (−0.148 − 0.988i)7-s + 2.27·8-s + (−0.166 + 0.288i)9-s + (−0.403 − 0.699i)10-s + (0.558 + 0.966i)11-s + (−0.651 + 1.12i)12-s + 1.51·13-s + (1.67 + 0.661i)14-s + 0.258·15-s + (−0.921 + 1.59i)16-s + (−0.161 − 0.280i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.306 - 0.951i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.306 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.306 - 0.951i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.306 - 0.951i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.655822 + 0.477751i\)
\(L(\frac12)\) \(\approx\) \(0.655822 + 0.477751i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 + 2.59i)T \)
5 \( 1 + (2.5 - 4.33i)T \)
7 \( 1 + (2.74 + 18.3i)T \)
good2 \( 1 + (2.55 - 4.42i)T + (-4 - 6.92i)T^{2} \)
11 \( 1 + (-20.3 - 35.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 70.9T + 2.19e3T^{2} \)
17 \( 1 + (11.3 + 19.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (38.8 - 67.2i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-45.0 + 77.9i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 213.T + 2.43e4T^{2} \)
31 \( 1 + (-87.8 - 152. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-177. + 307. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 249.T + 6.89e4T^{2} \)
43 \( 1 - 297.T + 7.95e4T^{2} \)
47 \( 1 + (84.3 - 146. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-51.2 - 88.7i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-11.7 - 20.3i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-437. + 757. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-405. - 702. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 632.T + 3.57e5T^{2} \)
73 \( 1 + (457. + 792. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-260. + 451. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.02e3T + 5.71e5T^{2} \)
89 \( 1 + (501. - 869. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 639.T + 9.12e5T^{2} \)
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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

−13.98857519355722129788003855058, −12.67762503919172022768112715873, −10.98910709246734375377453258581, −10.10060788377543059376898257189, −8.787091894390631408304701104956, −7.73498776637320124310202579658, −6.82274890328155027410734502819, −6.14751925437580517410567451073, −4.35415658296865154323022544627, −0.979104398831341465789453249294, 0.972257530760754125596569773211, 2.93094378615670198687058174549, 4.19636381504464884056042703456, 6.05485024681368056872223104352, 8.476082772207164724263595634553, 8.807847311565082466722173521720, 9.911958919495778871766525746861, 11.28742126006051647533668663796, 11.43583596492004851316514951761, 12.68723232292181868542790305846

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