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Properties

Label 2-3150-21.5-c1-0-12
Degree $2$
Conductor $3150$
Sign $-0.153 - 0.988i$
Analytic cond. $25.1528$
Root an. cond. $5.01526$
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  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.397 + 2.61i)7-s − 0.999i·8-s + (0.429 − 0.248i)11-s + 2.74i·13-s + (1.65 − 2.06i)14-s + (−0.5 + 0.866i)16-s + (1.82 + 3.16i)17-s + (3.12 + 1.80i)19-s − 0.496·22-s + (5.56 + 3.21i)23-s + (1.37 − 2.37i)26-s + (−2.46 + 0.963i)28-s − 8.87i·29-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.150 + 0.988i)7-s − 0.353i·8-s + (0.129 − 0.0748i)11-s + 0.761i·13-s + (0.441 − 0.552i)14-s + (−0.125 + 0.216i)16-s + (0.443 + 0.768i)17-s + (0.716 + 0.413i)19-s − 0.105·22-s + (1.16 + 0.669i)23-s + (0.269 − 0.466i)26-s + (−0.465 + 0.182i)28-s − 1.64i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.153 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.153 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.153 - 0.988i$
Analytic conductor: \(25.1528\)
Root analytic conductor: \(5.01526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3150} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3150,\ (\ :1/2),\ -0.153 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.049805567\)
\(L(\frac12)\) \(\approx\) \(1.049805567\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (0.397 - 2.61i)T \)
good11 \( 1 + (-0.429 + 0.248i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.74iT - 13T^{2} \)
17 \( 1 + (-1.82 - 3.16i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.12 - 1.80i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.56 - 3.21i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.87iT - 29T^{2} \)
31 \( 1 + (6.90 - 3.98i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.14 + 1.98i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.22T + 41T^{2} \)
43 \( 1 + 2.22T + 43T^{2} \)
47 \( 1 + (3.27 - 5.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.72 + 3.88i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.05 + 5.28i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.24 - 1.87i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.08 - 7.08i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + (11.3 - 6.53i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.44 + 7.69i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.79T + 83T^{2} \)
89 \( 1 + (-0.743 + 1.28i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 9.05iT - 97T^{2} \)
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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

−8.955262268684977663823648525506, −8.309343872894678586497336693489, −7.52178035125793698749679069075, −6.72891473886944013231475569006, −5.87578143833326442264491092416, −5.18781963114596749589894526527, −3.98012893484767194450886882437, −3.20075313425383336246078700646, −2.20603309880635324115088330045, −1.29690872683334398231679634402, 0.43925873302295285400800697193, 1.38525245532892692076482261284, 2.85164054790960044913961527319, 3.59751702900812238358059073180, 4.85206976207797317623988209586, 5.36481578836441480316465087228, 6.47111209936796724519798958622, 7.21108098876823263716900927382, 7.51847485762069403646399319279, 8.491226925311972093819795256149

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