Properties

Label 2-30e2-100.23-c1-0-26
Degree $2$
Conductor $900$
Sign $0.942 + 0.334i$
Analytic cond. $7.18653$
Root an. cond. $2.68077$
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  + (0.213 − 1.39i)2-s + (−1.90 − 0.595i)4-s + (1.90 + 1.17i)5-s + (2.16 + 2.16i)7-s + (−1.23 + 2.54i)8-s + (2.04 − 2.40i)10-s + (−3.61 − 4.97i)11-s + (0.749 + 4.72i)13-s + (3.48 − 2.56i)14-s + (3.29 + 2.27i)16-s + (3.51 + 1.79i)17-s + (0.678 + 2.08i)19-s + (−2.93 − 3.37i)20-s + (−7.73 + 3.99i)22-s + (−0.374 + 2.36i)23-s + ⋯
L(s)  = 1  + (0.150 − 0.988i)2-s + (−0.954 − 0.297i)4-s + (0.850 + 0.525i)5-s + (0.817 + 0.817i)7-s + (−0.438 + 0.898i)8-s + (0.647 − 0.762i)10-s + (−1.09 − 1.50i)11-s + (0.207 + 1.31i)13-s + (0.931 − 0.685i)14-s + (0.822 + 0.568i)16-s + (0.852 + 0.434i)17-s + (0.155 + 0.479i)19-s + (−0.655 − 0.754i)20-s + (−1.64 + 0.852i)22-s + (−0.0781 + 0.493i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.942 + 0.334i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (523, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ 0.942 + 0.334i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81015 - 0.311325i\)
\(L(\frac12)\) \(\approx\) \(1.81015 - 0.311325i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.213 + 1.39i)T \)
3 \( 1 \)
5 \( 1 + (-1.90 - 1.17i)T \)
good7 \( 1 + (-2.16 - 2.16i)T + 7iT^{2} \)
11 \( 1 + (3.61 + 4.97i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.749 - 4.72i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-3.51 - 1.79i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-0.678 - 2.08i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.374 - 2.36i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (1.48 + 0.480i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-8.29 + 2.69i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-4.07 + 0.644i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (-5.32 - 3.86i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (8.72 - 8.72i)T - 43iT^{2} \)
47 \( 1 + (-3.47 + 1.77i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-5.54 + 2.82i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (7.51 + 5.46i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.715 - 0.519i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-1.52 + 2.98i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (5.96 + 1.93i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.906 - 0.143i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-2.98 + 9.17i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-6.49 - 3.31i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (0.440 + 0.606i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (7.49 + 14.7i)T + (-57.0 + 78.4i)T^{2} \)
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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

−10.11802103674572688480552728599, −9.453670277433350561674761333761, −8.500509693527912398790480486167, −7.910909086021980594454525977230, −6.15217810417298522146555442964, −5.69373328375882962839800496002, −4.73830314057303887329945314908, −3.36989837021267752598167538843, −2.48709498532910147672588618672, −1.45883775908046964418610106721, 0.960439501459766255171930136021, 2.69212998620400082975752705063, 4.31951277903398438477799639001, 5.05016288608265463282294478621, 5.58551003484206922869247025839, 6.84936823697857497011602515615, 7.68180491952461002725909065479, 8.154328875528721060265077035300, 9.246210698966387276688722642291, 10.22489597104581188439777942464

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