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Properties

Label 2-105-21.20-c3-0-30
Degree $2$
Conductor $105$
Sign $-0.341 - 0.940i$
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  − 5.54i·2-s + (3.35 − 3.96i)3-s − 22.7·4-s − 5·5-s + (−22.0 − 18.6i)6-s + (9.21 − 16.0i)7-s + 82.0i·8-s + (−4.49 − 26.6i)9-s + 27.7i·10-s + 18.7i·11-s + (−76.4 + 90.4i)12-s + 6.49i·13-s + (−89.1 − 51.1i)14-s + (−16.7 + 19.8i)15-s + 272.·16-s − 11.6·17-s + ⋯
L(s)  = 1  − 1.96i·2-s + (0.645 − 0.763i)3-s − 2.84·4-s − 0.447·5-s + (−1.49 − 1.26i)6-s + (0.497 − 0.867i)7-s + 3.62i·8-s + (−0.166 − 0.986i)9-s + 0.877i·10-s + 0.513i·11-s + (−1.83 + 2.17i)12-s + 0.138i·13-s + (−1.70 − 0.976i)14-s + (−0.288 + 0.341i)15-s + 4.26·16-s − 0.166·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.341 - 0.940i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.341 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.341 - 0.940i$
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: \((2,\ 105,\ (\ :3/2),\ -0.341 - 0.940i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.724425 + 1.03357i\)
\(L(\frac12)\) \(\approx\) \(0.724425 + 1.03357i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.35 + 3.96i)T \)
5 \( 1 + 5T \)
7 \( 1 + (-9.21 + 16.0i)T \)
good2 \( 1 + 5.54iT - 8T^{2} \)
11 \( 1 - 18.7iT - 1.33e3T^{2} \)
13 \( 1 - 6.49iT - 2.19e3T^{2} \)
17 \( 1 + 11.6T + 4.91e3T^{2} \)
19 \( 1 + 123. iT - 6.85e3T^{2} \)
23 \( 1 + 5.61iT - 1.21e4T^{2} \)
29 \( 1 - 174. iT - 2.43e4T^{2} \)
31 \( 1 + 213. iT - 2.97e4T^{2} \)
37 \( 1 + 176.T + 5.06e4T^{2} \)
41 \( 1 + 187.T + 6.89e4T^{2} \)
43 \( 1 - 164.T + 7.95e4T^{2} \)
47 \( 1 - 440.T + 1.03e5T^{2} \)
53 \( 1 + 144. iT - 1.48e5T^{2} \)
59 \( 1 - 49.8T + 2.05e5T^{2} \)
61 \( 1 + 308. iT - 2.26e5T^{2} \)
67 \( 1 - 97.4T + 3.00e5T^{2} \)
71 \( 1 + 491. iT - 3.57e5T^{2} \)
73 \( 1 - 787. iT - 3.89e5T^{2} \)
79 \( 1 - 824.T + 4.93e5T^{2} \)
83 \( 1 - 410.T + 5.71e5T^{2} \)
89 \( 1 - 1.61e3T + 7.04e5T^{2} \)
97 \( 1 + 747. iT - 9.12e5T^{2} \)
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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

−12.47693297522295276075490897816, −11.53977242201553623855269873404, −10.70017590973495678516746999893, −9.442215370103191302265589258669, −8.506347990652345338875669918249, −7.30212210484611740965794362957, −4.72138222017055928974302911099, −3.56562061790244951714061386142, −2.14370714811609736173771814531, −0.69670298248386828191249643030, 3.72341395109778374259403634336, 4.98221494828041473124458109169, 5.98073610209953907237755055054, 7.62687960587178986904124467935, 8.393444263279199375512387700740, 9.086034361761343659161381018220, 10.35315712513223645453654780942, 12.23607846895265315461163828115, 13.64811917459526566661238101605, 14.39246573847555474861391572046

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