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Properties

Label 2-105-35.2-c2-0-4
Degree $2$
Conductor $105$
Sign $0.337 - 0.941i$
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  + (0.901 − 0.241i)2-s + (−1.67 − 0.448i)3-s + (−2.71 + 1.56i)4-s + (2.44 + 4.36i)5-s − 1.61·6-s + (5.41 + 4.44i)7-s + (−4.70 + 4.70i)8-s + (2.59 + 1.50i)9-s + (3.25 + 3.34i)10-s + (3.17 + 5.49i)11-s + (5.23 − 1.40i)12-s + (−5.74 + 5.74i)13-s + (5.94 + 2.69i)14-s + (−2.13 − 8.39i)15-s + (3.15 − 5.46i)16-s + (0.226 − 0.846i)17-s + ⋯
L(s)  = 1  + (0.450 − 0.120i)2-s + (−0.557 − 0.149i)3-s + (−0.677 + 0.391i)4-s + (0.488 + 0.872i)5-s − 0.269·6-s + (0.773 + 0.634i)7-s + (−0.587 + 0.587i)8-s + (0.288 + 0.166i)9-s + (0.325 + 0.334i)10-s + (0.288 + 0.499i)11-s + (0.436 − 0.116i)12-s + (−0.442 + 0.442i)13-s + (0.424 + 0.192i)14-s + (−0.142 − 0.559i)15-s + (0.197 − 0.341i)16-s + (0.0133 − 0.0498i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.337 - 0.941i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.337 - 0.941i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.05103 + 0.739385i\)
\(L(\frac12)\) \(\approx\) \(1.05103 + 0.739385i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.67 + 0.448i)T \)
5 \( 1 + (-2.44 - 4.36i)T \)
7 \( 1 + (-5.41 - 4.44i)T \)
good2 \( 1 + (-0.901 + 0.241i)T + (3.46 - 2i)T^{2} \)
11 \( 1 + (-3.17 - 5.49i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (5.74 - 5.74i)T - 169iT^{2} \)
17 \( 1 + (-0.226 + 0.846i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (-5.02 - 2.90i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (9.99 + 37.3i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 + 28.7iT - 841T^{2} \)
31 \( 1 + (-30.5 - 52.9i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (11.3 - 3.03i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 21.6T + 1.68e3T^{2} \)
43 \( 1 + (-45.3 + 45.3i)T - 1.84e3iT^{2} \)
47 \( 1 + (-49.7 + 13.3i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (43.5 + 11.6i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-50.4 + 29.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (14.0 - 24.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-9.00 + 33.6i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 22.2T + 5.04e3T^{2} \)
73 \( 1 + (-22.4 - 6.02i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (-127. - 73.6i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (86.8 - 86.8i)T - 6.88e3iT^{2} \)
89 \( 1 + (47.6 + 27.5i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (114. + 114. i)T + 9.40e3iT^{2} \)
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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

−13.95312116363826211940524015335, −12.45312666454899419489167612443, −11.92869352829030589347358951367, −10.70641656515655257709365511047, −9.507039934669502680914777897316, −8.234974865548855625019022960126, −6.81220226001073277802337179790, −5.52021266504791685617662438036, −4.36384714898625798709651111826, −2.43607765451991780008499271147, 1.00093176933173677763395084011, 4.09982823789226325375019673203, 5.13336303292197074830085662109, 5.98304954974790835504205317487, 7.78873923411978451990309637484, 9.166834219642748849197714052555, 10.03135896927693760795470671641, 11.28973332907030985419471971478, 12.45014663291596744848649878768, 13.45857984373714683388381283747

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