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Properties

Label 2-3150-21.17-c1-0-7
Degree $2$
Conductor $3150$
Sign $0.100 - 0.994i$
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 + (−2.62 − 0.358i)7-s + 0.999i·8-s + (−2.59 − 1.5i)11-s − 2.44i·13-s + (2.44 − i)14-s + (−0.5 − 0.866i)16-s + (2.95 − 5.12i)17-s + (−5.12 + 2.95i)19-s + 3·22-s + (−3.67 + 2.12i)23-s + (1.22 + 2.12i)26-s + (−1.62 + 2.09i)28-s + 7.24i·29-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.990 − 0.135i)7-s + 0.353i·8-s + (−0.783 − 0.452i)11-s − 0.679i·13-s + (0.654 − 0.267i)14-s + (−0.125 − 0.216i)16-s + (0.717 − 1.24i)17-s + (−1.17 + 0.678i)19-s + 0.639·22-s + (−0.766 + 0.442i)23-s + (0.240 + 0.416i)26-s + (−0.306 + 0.395i)28-s + 1.34i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.100 - 0.994i$
Analytic conductor: \(25.1528\)
Root analytic conductor: \(5.01526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3150} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3150,\ (\ :1/2),\ 0.100 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5787764919\)
\(L(\frac12)\) \(\approx\) \(0.5787764919\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.62 + 0.358i)T \)
good11 \( 1 + (2.59 + 1.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.44iT - 13T^{2} \)
17 \( 1 + (-2.95 + 5.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.12 - 2.95i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.67 - 2.12i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.24iT - 29T^{2} \)
31 \( 1 + (7.86 + 4.54i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.121 + 0.210i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 11.8T + 41T^{2} \)
43 \( 1 - 0.242T + 43T^{2} \)
47 \( 1 + (-2.95 - 5.12i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.27 - 3.62i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.03 + 6.98i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.878 + 0.507i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5 - 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.75iT - 71T^{2} \)
73 \( 1 + (-1.24 - 0.717i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.37 - 2.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.63T + 83T^{2} \)
89 \( 1 + (-5.19 - 9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 13.5iT - 97T^{2} \)
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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

−8.963999523610793005419934787688, −7.934381338950786172514166853342, −7.54228171024679255469400081901, −6.70775477061188036023846313028, −5.74521764255956550861677182234, −5.47094845609024238215398754536, −4.09586541496072180862625666408, −3.16898121966913030585047533929, −2.31668643047815727870522719184, −0.792971381362917272931947788805, 0.29574091581702251528318757179, 1.92325686253028341240956056872, 2.58954535698109018311234781500, 3.73367452017986834486882804579, 4.36059444641613049764801519917, 5.67859243494536862018872342024, 6.29918796144538775317491725389, 7.12376792539545159599590697112, 7.82283273346788174181445636803, 8.684092342216333875566000649297

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