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Properties

Label 2-30e2-9.4-c1-0-15
Degree $2$
Conductor $900$
Sign $0.561 + 0.827i$
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  + (1.16 + 1.28i)3-s + (−2.49 − 4.32i)7-s + (−0.304 + 2.98i)9-s + (−1.99 − 3.46i)11-s + (0.771 − 1.33i)13-s + 6.99·17-s − 2.25·19-s + (2.66 − 8.23i)21-s + (3.89 − 6.75i)23-s + (−4.18 + 3.07i)27-s + (−3.08 − 5.33i)29-s + (0.271 − 0.470i)31-s + (2.12 − 6.58i)33-s + 6.25·37-s + (2.61 − 0.559i)39-s + ⋯
L(s)  = 1  + (0.670 + 0.742i)3-s + (−0.944 − 1.63i)7-s + (−0.101 + 0.994i)9-s + (−0.602 − 1.04i)11-s + (0.213 − 0.370i)13-s + 1.69·17-s − 0.517·19-s + (0.580 − 1.79i)21-s + (0.812 − 1.40i)23-s + (−0.806 + 0.591i)27-s + (−0.572 − 0.991i)29-s + (0.0487 − 0.0844i)31-s + (0.370 − 1.14i)33-s + 1.02·37-s + (0.418 − 0.0896i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.561 + 0.827i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ 0.561 + 0.827i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37362 - 0.727609i\)
\(L(\frac12)\) \(\approx\) \(1.37362 - 0.727609i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.16 - 1.28i)T \)
5 \( 1 \)
good7 \( 1 + (2.49 + 4.32i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.99 + 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.771 + 1.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 6.99T + 17T^{2} \)
19 \( 1 + 2.25T + 19T^{2} \)
23 \( 1 + (-3.89 + 6.75i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.08 + 5.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.271 + 0.470i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.25T + 37T^{2} \)
41 \( 1 + (0.0979 - 0.169i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.0431 + 0.0747i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.91 + 3.31i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 4.19T + 53T^{2} \)
59 \( 1 + (3.51 - 6.08i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.45 - 2.51i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.48 + 7.76i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.79T + 71T^{2} \)
73 \( 1 - 2.28T + 73T^{2} \)
79 \( 1 + (-6.32 - 10.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.98 + 12.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + (-4.66 - 8.08i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.09092127028513603057036537979, −9.315423553970489658986223802280, −8.159037762070653665920433689617, −7.72275349334640604289279244766, −6.57594267585491468264469806615, −5.55401381778682598771003711505, −4.35940572693162551554700299213, −3.53319262511845990786755417986, −2.83967304169391600802534426431, −0.69208547089590597002966602674, 1.67840495869674384509511319092, 2.74649974093181215221891046517, 3.50245550944784135073818288144, 5.19614850186789582540291474849, 5.96071778397806278386559507868, 6.90512400809514724912413330027, 7.72565909110127696416381774279, 8.568397888334474112219994386098, 9.557069332278276555381166551931, 9.677971498239231458036142535279

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