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Properties

Label 2-105-35.4-c3-0-6
Degree $2$
Conductor $105$
Sign $0.100 - 0.994i$
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  + (−0.761 + 0.439i)2-s + (−2.59 − 1.5i)3-s + (−3.61 + 6.25i)4-s + (0.571 − 11.1i)5-s + 2.63·6-s + (18.5 − 0.260i)7-s − 13.3i·8-s + (4.5 + 7.79i)9-s + (4.47 + 8.75i)10-s + (−32.2 + 55.8i)11-s + (18.7 − 10.8i)12-s + 73.6i·13-s + (−13.9 + 8.33i)14-s + (−18.2 + 28.1i)15-s + (−23.0 − 39.8i)16-s + (13.4 + 7.79i)17-s + ⋯
L(s)  = 1  + (−0.269 + 0.155i)2-s + (−0.499 − 0.288i)3-s + (−0.451 + 0.782i)4-s + (0.0511 − 0.998i)5-s + 0.179·6-s + (0.999 − 0.0140i)7-s − 0.591i·8-s + (0.166 + 0.288i)9-s + (0.141 + 0.276i)10-s + (−0.883 + 1.53i)11-s + (0.451 − 0.260i)12-s + 1.57i·13-s + (−0.266 + 0.159i)14-s + (−0.313 + 0.484i)15-s + (−0.359 − 0.623i)16-s + (0.192 + 0.111i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.100 - 0.994i$
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: \((2,\ 105,\ (\ :3/2),\ 0.100 - 0.994i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.679699 + 0.614648i\)
\(L(\frac12)\) \(\approx\) \(0.679699 + 0.614648i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.571 + 11.1i)T \)
7 \( 1 + (-18.5 + 0.260i)T \)
good2 \( 1 + (0.761 - 0.439i)T + (4 - 6.92i)T^{2} \)
11 \( 1 + (32.2 - 55.8i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 73.6iT - 2.19e3T^{2} \)
17 \( 1 + (-13.4 - 7.79i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-68.6 - 118. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-90.0 + 52.0i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 76.4T + 2.43e4T^{2} \)
31 \( 1 + (48.0 - 83.2i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (211. - 122. i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 28.9T + 6.89e4T^{2} \)
43 \( 1 + 47.5iT - 7.95e4T^{2} \)
47 \( 1 + (-100. + 58.1i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (7.99 + 4.61i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (52.5 - 90.9i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (95.6 + 165. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-436. - 252. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 1.08e3T + 3.57e5T^{2} \)
73 \( 1 + (398. + 229. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-8.94 - 15.4i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 691. iT - 5.71e5T^{2} \)
89 \( 1 + (-476. - 825. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 600. iT - 9.12e5T^{2} \)
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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

−13.33476729952416527105179042928, −12.26518312664536311988375843687, −11.90632428564452699324683300934, −10.19332439075171058166606722439, −9.033234581176676976317978604562, −7.992414515344892365547311057545, −7.11571540190506410419004293436, −5.13170155000377588436194775704, −4.34937250736568293846739021590, −1.64565098773474516067857802743, 0.63914599166602547648982828018, 2.99583531646100928291383579920, 5.12901298637466033434850935406, 5.78891427584118483771967898068, 7.54317615395648167589977985126, 8.756995403088257534212590850256, 10.17591858123962497116641609392, 10.91423807256849830941703468855, 11.36799278838204272759153405812, 13.29560652666310758293820157657

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