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Properties

Label 2-105-21.20-c1-0-7
Degree $2$
Conductor $105$
Sign $0.994 - 0.102i$
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  + 0.792i·2-s + (1.18 − 1.26i)3-s + 1.37·4-s − 5-s + (1 + 0.939i)6-s + (−2 − 1.73i)7-s + 2.67i·8-s + (−0.186 − 2.99i)9-s − 0.792i·10-s + 2.52i·11-s + (1.62 − 1.73i)12-s + 4.10i·13-s + (1.37 − 1.58i)14-s + (−1.18 + 1.26i)15-s + 0.627·16-s − 4.37·17-s + ⋯
L(s)  = 1  + 0.560i·2-s + (0.684 − 0.728i)3-s + 0.686·4-s − 0.447·5-s + (0.408 + 0.383i)6-s + (−0.755 − 0.654i)7-s + 0.944i·8-s + (−0.0620 − 0.998i)9-s − 0.250i·10-s + 0.761i·11-s + (0.469 − 0.499i)12-s + 1.13i·13-s + (0.366 − 0.423i)14-s + (−0.306 + 0.325i)15-s + 0.156·16-s − 1.06·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.102i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.994 - 0.102i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ 0.994 - 0.102i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25188 + 0.0643513i\)
\(L(\frac12)\) \(\approx\) \(1.25188 + 0.0643513i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.18 + 1.26i)T \)
5 \( 1 + T \)
7 \( 1 + (2 + 1.73i)T \)
good2 \( 1 - 0.792iT - 2T^{2} \)
11 \( 1 - 2.52iT - 11T^{2} \)
13 \( 1 - 4.10iT - 13T^{2} \)
17 \( 1 + 4.37T + 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 + 8.51iT - 23T^{2} \)
29 \( 1 + 0.939iT - 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + 6.74T + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 4.74T + 43T^{2} \)
47 \( 1 - 1.62T + 47T^{2} \)
53 \( 1 - 1.87iT - 53T^{2} \)
59 \( 1 - 8.74T + 59T^{2} \)
61 \( 1 + 6.92iT - 61T^{2} \)
67 \( 1 - 4.74T + 67T^{2} \)
71 \( 1 + 0.294iT - 71T^{2} \)
73 \( 1 + 6.92iT - 73T^{2} \)
79 \( 1 + 2.37T + 79T^{2} \)
83 \( 1 - 17.4T + 83T^{2} \)
89 \( 1 + 14.7T + 89T^{2} \)
97 \( 1 - 11.0iT - 97T^{2} \)
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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

−13.99570984283383166099339342286, −12.75878285842498711771770319624, −11.94161070205739558113743451461, −10.64810091764996883318875438457, −9.215089763845090932419329154997, −8.006822978748833543325066029028, −6.96757189425657258031323620066, −6.44647856948681434841951246842, −4.10732523641752119583992586577, −2.30252198144227541595869508941, 2.70541935217402072684823201972, 3.62559242695775115179724846797, 5.57204777826069712473561720948, 7.16300210209892708671766607057, 8.533075451032723520702541435120, 9.553473933614660073072587010671, 10.67926379830545023237099614021, 11.43274926425440731678118307626, 12.70316754624541437728065584222, 13.58288047496936766160331260165

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