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Properties

Label 2-105-35.9-c1-0-4
Degree $2$
Conductor $105$
Sign $0.460 + 0.887i$
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  + (−0.248 − 0.143i)2-s + (−0.866 + 0.5i)3-s + (−0.958 − 1.66i)4-s + (1.47 − 1.68i)5-s + 0.287·6-s + (1.11 − 2.39i)7-s + 1.12i·8-s + (0.499 − 0.866i)9-s + (−0.608 + 0.206i)10-s + (1.66 + 2.88i)11-s + (1.66 + 0.958i)12-s − 4.54i·13-s + (−0.622 + 0.436i)14-s + (−0.437 + 2.19i)15-s + (−1.75 + 3.04i)16-s + (−4.80 + 2.77i)17-s + ⋯
L(s)  = 1  + (−0.175 − 0.101i)2-s + (−0.499 + 0.288i)3-s + (−0.479 − 0.830i)4-s + (0.659 − 0.751i)5-s + 0.117·6-s + (0.421 − 0.906i)7-s + 0.397i·8-s + (0.166 − 0.288i)9-s + (−0.192 + 0.0652i)10-s + (0.502 + 0.869i)11-s + (0.479 + 0.276i)12-s − 1.26i·13-s + (−0.166 + 0.116i)14-s + (−0.112 + 0.566i)15-s + (−0.438 + 0.760i)16-s + (−1.16 + 0.672i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.460 + 0.887i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ 0.460 + 0.887i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.710348 - 0.431606i\)
\(L(\frac12)\) \(\approx\) \(0.710348 - 0.431606i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (-1.47 + 1.68i)T \)
7 \( 1 + (-1.11 + 2.39i)T \)
good2 \( 1 + (0.248 + 0.143i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (-1.66 - 2.88i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.54iT - 13T^{2} \)
17 \( 1 + (4.80 - 2.77i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.828 - 1.43i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.61 - 3.81i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.118T + 29T^{2} \)
31 \( 1 + (-3.13 - 5.42i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.71 - 3.87i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.0701T + 41T^{2} \)
43 \( 1 - 2.92iT - 43T^{2} \)
47 \( 1 + (5.53 + 3.19i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.640 + 0.369i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.815 + 1.41i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.65 + 6.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.62 - 1.51i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.77T + 71T^{2} \)
73 \( 1 + (-2.03 + 1.17i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.97 - 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.22iT - 83T^{2} \)
89 \( 1 + (6.50 - 11.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.04iT - 97T^{2} \)
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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

−13.45949289916608773904435950269, −12.76663511377181987604362452418, −11.18139891463727652971560979569, −10.30173605340592590996763825993, −9.548449422175279199505467452026, −8.349891259719555913449667102454, −6.59499797362631149300492477838, −5.26626274865403919145785270553, −4.42211254221182107033363341092, −1.30292915341268219899589273766, 2.59554988575533808721808078263, 4.58024174991662823391374177435, 6.18629582279236059308492106344, 7.11326545485791928647907164454, 8.698813853104137263467874786281, 9.358467891376243462824688487571, 11.15034196558202533489724680817, 11.66164414466802190978513853698, 13.00852231533102368530388496561, 13.80180673863690511955683574666

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