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Properties

Label 2-105-21.20-c3-0-23
Degree $2$
Conductor $105$
Sign $0.647 + 0.761i$
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  − 1.39i·2-s + (5.17 − 0.416i)3-s + 6.05·4-s + 5·5-s + (−0.580 − 7.21i)6-s + (−10.8 − 15.0i)7-s − 19.5i·8-s + (26.6 − 4.31i)9-s − 6.96i·10-s + 67.7i·11-s + (31.3 − 2.52i)12-s − 2.57i·13-s + (−20.9 + 15.0i)14-s + (25.8 − 2.08i)15-s + 21.1·16-s − 62.5·17-s + ⋯
L(s)  = 1  − 0.492i·2-s + (0.996 − 0.0801i)3-s + 0.757·4-s + 0.447·5-s + (−0.0394 − 0.491i)6-s + (−0.584 − 0.811i)7-s − 0.865i·8-s + (0.987 − 0.159i)9-s − 0.220i·10-s + 1.85i·11-s + (0.754 − 0.0606i)12-s − 0.0550i·13-s + (−0.399 + 0.288i)14-s + (0.445 − 0.0358i)15-s + 0.330·16-s − 0.892·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.647 + 0.761i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.647 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.647 + 0.761i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.647 + 0.761i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.33421 - 1.07941i\)
\(L(\frac12)\) \(\approx\) \(2.33421 - 1.07941i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.17 + 0.416i)T \)
5 \( 1 - 5T \)
7 \( 1 + (10.8 + 15.0i)T \)
good2 \( 1 + 1.39iT - 8T^{2} \)
11 \( 1 - 67.7iT - 1.33e3T^{2} \)
13 \( 1 + 2.57iT - 2.19e3T^{2} \)
17 \( 1 + 62.5T + 4.91e3T^{2} \)
19 \( 1 + 134. iT - 6.85e3T^{2} \)
23 \( 1 - 58.4iT - 1.21e4T^{2} \)
29 \( 1 - 91.3iT - 2.43e4T^{2} \)
31 \( 1 - 182. iT - 2.97e4T^{2} \)
37 \( 1 + 340.T + 5.06e4T^{2} \)
41 \( 1 - 67.1T + 6.89e4T^{2} \)
43 \( 1 + 88.7T + 7.95e4T^{2} \)
47 \( 1 - 157.T + 1.03e5T^{2} \)
53 \( 1 - 458. iT - 1.48e5T^{2} \)
59 \( 1 + 589.T + 2.05e5T^{2} \)
61 \( 1 - 431. iT - 2.26e5T^{2} \)
67 \( 1 - 809.T + 3.00e5T^{2} \)
71 \( 1 + 80.5iT - 3.57e5T^{2} \)
73 \( 1 + 406. iT - 3.89e5T^{2} \)
79 \( 1 + 657.T + 4.93e5T^{2} \)
83 \( 1 + 649.T + 5.71e5T^{2} \)
89 \( 1 + 600.T + 7.04e5T^{2} \)
97 \( 1 + 813. iT - 9.12e5T^{2} \)
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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

−13.04782350472242941549583732236, −12.35949495114264600521566386995, −10.78164331343382330028053881002, −9.960866760115645747756602999586, −9.087857934179414504772182521675, −7.16807000900737744145828061454, −6.92584146797199424404760752020, −4.46463428206978038061987187241, −2.95616502020581607787464122416, −1.70955742564295790335502950974, 2.17497050147555072943752508376, 3.41069756901137959965026753322, 5.70572171123946920122650053285, 6.55447668671597264417319303723, 8.105715815937522948666142104677, 8.789776500569231147208825756412, 10.08160790408818258117595739925, 11.27669806314952793771111148258, 12.55722779582022126258041952342, 13.70157770773803410635807712629

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