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Properties

Label 2-105-21.17-c1-0-0
Degree $2$
Conductor $105$
Sign $0.227 - 0.973i$
Analytic cond. $0.838429$
Root an. cond. $0.915657$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.01 + 1.16i)2-s + (−1.21 − 1.23i)3-s + (1.71 − 2.97i)4-s + (0.5 + 0.866i)5-s + (3.89 + 1.07i)6-s + (1.11 + 2.39i)7-s + 3.33i·8-s + (−0.0404 + 2.99i)9-s + (−2.01 − 1.16i)10-s + (2.42 + 1.39i)11-s + (−5.75 + 1.50i)12-s + 3.20i·13-s + (−5.04 − 3.53i)14-s + (0.459 − 1.66i)15-s + (−0.459 − 0.795i)16-s + (−0.440 + 0.763i)17-s + ⋯
L(s)  = 1  + (−1.42 + 0.824i)2-s + (−0.702 − 0.711i)3-s + (0.858 − 1.48i)4-s + (0.223 + 0.387i)5-s + (1.58 + 0.437i)6-s + (0.422 + 0.906i)7-s + 1.18i·8-s + (−0.0134 + 0.999i)9-s + (−0.638 − 0.368i)10-s + (0.729 + 0.421i)11-s + (−1.66 + 0.433i)12-s + 0.888i·13-s + (−1.34 − 0.946i)14-s + (0.118 − 0.431i)15-s + (−0.114 − 0.198i)16-s + (−0.106 + 0.185i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.227 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.227 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.227 - 0.973i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ 0.227 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.357162 + 0.283329i\)
\(L(\frac12)\) \(\approx\) \(0.357162 + 0.283329i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.21 + 1.23i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-1.11 - 2.39i)T \)
good2 \( 1 + (2.01 - 1.16i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (-2.42 - 1.39i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.20iT - 13T^{2} \)
17 \( 1 + (0.440 - 0.763i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.90 + 1.09i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.53 + 3.77i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.15iT - 29T^{2} \)
31 \( 1 + (7.62 + 4.40i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.203 + 0.352i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.55T + 41T^{2} \)
43 \( 1 + 0.118T + 43T^{2} \)
47 \( 1 + (1.31 + 2.27i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.46 - 3.73i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.04 + 3.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.7 + 6.17i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.802 + 1.38i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.25iT - 71T^{2} \)
73 \( 1 + (-0.192 - 0.110i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.56 - 2.71i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.666T + 83T^{2} \)
89 \( 1 + (0.437 + 0.757i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.37iT - 97T^{2} \)
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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

−14.36963016660576444298237501847, −12.79406504200886615258345763439, −11.60334193393532182014587318765, −10.75680932029630261730227354968, −9.379909957346089332787866079307, −8.556819772962629444442371617431, −7.16518846903497013300984959131, −6.59157549360708681307352304586, −5.29668981685163018397146649296, −1.76990431563914865496509141169, 1.00903400038061717024357276711, 3.58139211027403480765315332422, 5.32056021840744485747838014687, 7.14335698229331352861591282516, 8.494650348796009933501665538266, 9.510810936704514405740974871645, 10.31778020255620935606129835219, 11.18171868532410412064445070943, 11.87467397974010327926292399056, 13.25198332920463506099148546299

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