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Properties

Label 2-105-35.23-c2-0-15
Degree $2$
Conductor $105$
Sign $-0.182 - 0.983i$
Analytic cond. $2.86104$
Root an. cond. $1.69146$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.945 − 3.52i)2-s + (0.448 − 1.67i)3-s + (−8.09 + 4.67i)4-s + (−4.69 − 1.71i)5-s − 6.32·6-s + (1.65 + 6.80i)7-s + (13.8 + 13.8i)8-s + (−2.59 − 1.50i)9-s + (−1.60 + 18.1i)10-s + (−7.38 − 12.7i)11-s + (4.18 + 15.6i)12-s + (−7.90 − 7.90i)13-s + (22.4 − 12.2i)14-s + (−4.97 + 7.08i)15-s + (16.9 − 29.3i)16-s + (8.26 + 2.21i)17-s + ⋯
L(s)  = 1  + (−0.472 − 1.76i)2-s + (0.149 − 0.557i)3-s + (−2.02 + 1.16i)4-s + (−0.939 − 0.342i)5-s − 1.05·6-s + (0.236 + 0.971i)7-s + (1.72 + 1.72i)8-s + (−0.288 − 0.166i)9-s + (−0.160 + 1.81i)10-s + (−0.671 − 1.16i)11-s + (0.349 + 1.30i)12-s + (−0.607 − 0.607i)13-s + (1.60 − 0.875i)14-s + (−0.331 + 0.472i)15-s + (1.06 − 1.83i)16-s + (0.485 + 0.130i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.182 - 0.983i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.182 - 0.983i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ -0.182 - 0.983i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.269374 + 0.324019i\)
\(L(\frac12)\) \(\approx\) \(0.269374 + 0.324019i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.448 + 1.67i)T \)
5 \( 1 + (4.69 + 1.71i)T \)
7 \( 1 + (-1.65 - 6.80i)T \)
good2 \( 1 + (0.945 + 3.52i)T + (-3.46 + 2i)T^{2} \)
11 \( 1 + (7.38 + 12.7i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (7.90 + 7.90i)T + 169iT^{2} \)
17 \( 1 + (-8.26 - 2.21i)T + (250. + 144.5i)T^{2} \)
19 \( 1 + (7.14 + 4.12i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (18.1 - 4.87i)T + (458. - 264.5i)T^{2} \)
29 \( 1 + 5.44iT - 841T^{2} \)
31 \( 1 + (26.5 + 45.9i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (9.25 + 34.5i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + 40.8T + 1.68e3T^{2} \)
43 \( 1 + (-53.7 - 53.7i)T + 1.84e3iT^{2} \)
47 \( 1 + (10.7 + 40.2i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-14.8 + 55.5i)T + (-2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-14.5 + 8.38i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-46.7 + 80.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-105. - 28.2i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 51.0T + 5.04e3T^{2} \)
73 \( 1 + (11.3 - 42.2i)T + (-4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-24.7 - 14.3i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (34.5 + 34.5i)T + 6.88e3iT^{2} \)
89 \( 1 + (45.7 + 26.4i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (33.9 - 33.9i)T - 9.40e3iT^{2} \)
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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

−12.55736358610227069791212301552, −11.64877832582355980004518398474, −11.01271705699102734090109984939, −9.613219368658532645580432168613, −8.431890448328813249303603102926, −7.935015265292729344732472406980, −5.38577334530674620931818406000, −3.62961645753425171228241604196, −2.36296050243390080998296587011, −0.35700717422613458661961511360, 4.12727624596326536204755158978, 5.05855387009776587809040568701, 6.89988580093579984445775037497, 7.50699121584288014419952865809, 8.469227228191429391907391243533, 9.824235780165380718096715985193, 10.60402223022966671687687147748, 12.33169976426301237601936734184, 13.94349091982043224601711465890, 14.58644931876536167385369023500

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