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Properties

Label 2-105-7.2-c3-0-12
Degree $2$
Conductor $105$
Sign $0.165 + 0.986i$
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  + (−1.18 + 2.05i)2-s + (−1.5 − 2.59i)3-s + (1.18 + 2.05i)4-s + (2.5 − 4.33i)5-s + 7.11·6-s + (−10.8 − 15.0i)7-s − 24.6·8-s + (−4.5 + 7.79i)9-s + (5.93 + 10.2i)10-s + (−9.32 − 16.1i)11-s + (3.55 − 6.15i)12-s + 12.9·13-s + (43.7 − 4.51i)14-s − 15.0·15-s + (19.7 − 34.1i)16-s + (−59.6 − 103. i)17-s + ⋯
L(s)  = 1  + (−0.419 + 0.726i)2-s + (−0.288 − 0.499i)3-s + (0.148 + 0.256i)4-s + (0.223 − 0.387i)5-s + 0.484·6-s + (−0.586 − 0.810i)7-s − 1.08·8-s + (−0.166 + 0.288i)9-s + (0.187 + 0.324i)10-s + (−0.255 − 0.442i)11-s + (0.0854 − 0.148i)12-s + 0.275·13-s + (0.834 − 0.0862i)14-s − 0.258·15-s + (0.308 − 0.533i)16-s + (−0.850 − 1.47i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.165 + 0.986i$
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: \((2,\ 105,\ (\ :3/2),\ 0.165 + 0.986i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.552831 - 0.467754i\)
\(L(\frac12)\) \(\approx\) \(0.552831 - 0.467754i\)
\(L(\frac{5}{2})\) 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 \( 1 + (10.8 + 15.0i)T \)
good2 \( 1 + (1.18 - 2.05i)T + (-4 - 6.92i)T^{2} \)
11 \( 1 + (9.32 + 16.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 12.9T + 2.19e3T^{2} \)
17 \( 1 + (59.6 + 103. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-74.4 + 128. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-11.1 + 19.3i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 147.T + 2.43e4T^{2} \)
31 \( 1 + (-167. - 289. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-59.0 + 102. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 505.T + 6.89e4T^{2} \)
43 \( 1 + 220.T + 7.95e4T^{2} \)
47 \( 1 + (-17.2 + 29.8i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-83.8 - 145. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-282. - 489. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-215. + 372. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (435. + 753. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 552.T + 3.57e5T^{2} \)
73 \( 1 + (-312. - 541. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (118. - 205. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 489.T + 5.71e5T^{2} \)
89 \( 1 + (-251. + 435. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 814.T + 9.12e5T^{2} \)
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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

−13.27853327201731615845251376482, −12.01062288819839586890283764821, −11.04277097009044676281562295634, −9.509470311984310103808660372409, −8.533861508640807001391980113419, −7.19513031129465397001551837540, −6.66076720933097572399551424593, −5.11531840633674703752560565020, −3.01996160889091424919029683794, −0.45653746705348151244030522429, 1.98833265554828714979898295007, 3.53795278747074475669565950385, 5.61117764419933273999932628686, 6.42809968161535181136580780102, 8.404110507103481035767869708915, 9.706333174286520869247707445810, 10.15727947412703708618925030489, 11.31098089318075748067001375850, 12.12975632610375440145905121204, 13.28478361855886811339244798081

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