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Properties

Label 2-2900-145.12-c1-0-10
Degree $2$
Conductor $2900$
Sign $-0.583 - 0.812i$
Analytic cond. $23.1566$
Root an. cond. $4.81213$
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  + 1.33·3-s + (0.751 + 0.751i)7-s − 1.22·9-s + (2.26 + 2.26i)11-s + (−2.06 − 2.06i)13-s + 3.91i·17-s + (−3.26 + 3.26i)19-s + (0.999 + 0.999i)21-s + (−3.58 + 3.58i)23-s − 5.62·27-s + (−2.97 + 4.49i)29-s + (−3.77 − 3.77i)31-s + (3.00 + 3.00i)33-s + 0.947·37-s + (−2.74 − 2.74i)39-s + ⋯
L(s)  = 1  + 0.768·3-s + (0.283 + 0.283i)7-s − 0.409·9-s + (0.681 + 0.681i)11-s + (−0.572 − 0.572i)13-s + 0.949i·17-s + (−0.748 + 0.748i)19-s + (0.218 + 0.218i)21-s + (−0.748 + 0.748i)23-s − 1.08·27-s + (−0.552 + 0.833i)29-s + (−0.678 − 0.678i)31-s + (0.523 + 0.523i)33-s + 0.155·37-s + (−0.439 − 0.439i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.583 - 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.583 - 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2900\)    =    \(2^{2} \cdot 5^{2} \cdot 29\)
Sign: $-0.583 - 0.812i$
Analytic conductor: \(23.1566\)
Root analytic conductor: \(4.81213\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2900} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2900,\ (\ :1/2),\ -0.583 - 0.812i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.327348256\)
\(L(\frac12)\) \(\approx\) \(1.327348256\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
29 \( 1 + (2.97 - 4.49i)T \)
good3 \( 1 - 1.33T + 3T^{2} \)
7 \( 1 + (-0.751 - 0.751i)T + 7iT^{2} \)
11 \( 1 + (-2.26 - 2.26i)T + 11iT^{2} \)
13 \( 1 + (2.06 + 2.06i)T + 13iT^{2} \)
17 \( 1 - 3.91iT - 17T^{2} \)
19 \( 1 + (3.26 - 3.26i)T - 19iT^{2} \)
23 \( 1 + (3.58 - 3.58i)T - 23iT^{2} \)
31 \( 1 + (3.77 + 3.77i)T + 31iT^{2} \)
37 \( 1 - 0.947T + 37T^{2} \)
41 \( 1 + (2.28 - 2.28i)T - 41iT^{2} \)
43 \( 1 + 10.3T + 43T^{2} \)
47 \( 1 - 11.4T + 47T^{2} \)
53 \( 1 + (3.18 - 3.18i)T - 53iT^{2} \)
59 \( 1 + 2.80iT - 59T^{2} \)
61 \( 1 + (-6.98 - 6.98i)T + 61iT^{2} \)
67 \( 1 + (-8.83 + 8.83i)T - 67iT^{2} \)
71 \( 1 - 4.22iT - 71T^{2} \)
73 \( 1 - 9.15iT - 73T^{2} \)
79 \( 1 + (11.2 - 11.2i)T - 79iT^{2} \)
83 \( 1 + (-11.4 + 11.4i)T - 83iT^{2} \)
89 \( 1 + (5.23 - 5.23i)T - 89iT^{2} \)
97 \( 1 + 9.97T + 97T^{2} \)
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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

−8.911488717356006734588570561112, −8.286319455210190823751234257002, −7.73114005842386355774309735105, −6.86837108811873427481240739997, −5.89272893610637731895200514560, −5.27065013053527978548701521183, −4.06212786565732705569975279054, −3.52780717701532079573717376828, −2.32938518669462915942286627602, −1.67270697997409834771840390608, 0.34222080940070788149987447664, 1.91954802279189425672816273208, 2.71336801335734842244081909714, 3.67850582499730034042568429000, 4.44466558950078199461263132671, 5.37139211980196727484872549417, 6.34546069179537440243269053004, 7.05163271227213659510628111580, 7.85492715390466627844880678646, 8.616879492041106974701380743913

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