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Properties

Label 2-1450-145.12-c1-0-12
Degree $2$
Conductor $1450$
Sign $-0.635 - 0.771i$
Analytic cond. $11.5783$
Root an. cond. $3.40269$
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  + i·2-s − 0.525·3-s − 4-s − 0.525i·6-s + (1.99 + 1.99i)7-s − i·8-s − 2.72·9-s + (4.00 + 4.00i)11-s + 0.525·12-s + (1.85 + 1.85i)13-s + (−1.99 + 1.99i)14-s + 16-s − 4.25i·17-s − 2.72i·18-s + (−0.197 + 0.197i)19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.303·3-s − 0.5·4-s − 0.214i·6-s + (0.754 + 0.754i)7-s − 0.353i·8-s − 0.907·9-s + (1.20 + 1.20i)11-s + 0.151·12-s + (0.513 + 0.513i)13-s + (−0.533 + 0.533i)14-s + 0.250·16-s − 1.03i·17-s − 0.641i·18-s + (−0.0452 + 0.0452i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 - 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.635 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1450\)    =    \(2 \cdot 5^{2} \cdot 29\)
Sign: $-0.635 - 0.771i$
Analytic conductor: \(11.5783\)
Root analytic conductor: \(3.40269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1450} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1450,\ (\ :1/2),\ -0.635 - 0.771i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.417738729\)
\(L(\frac12)\) \(\approx\) \(1.417738729\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 \)
29 \( 1 + (-0.248 - 5.37i)T \)
good3 \( 1 + 0.525T + 3T^{2} \)
7 \( 1 + (-1.99 - 1.99i)T + 7iT^{2} \)
11 \( 1 + (-4.00 - 4.00i)T + 11iT^{2} \)
13 \( 1 + (-1.85 - 1.85i)T + 13iT^{2} \)
17 \( 1 + 4.25iT - 17T^{2} \)
19 \( 1 + (0.197 - 0.197i)T - 19iT^{2} \)
23 \( 1 + (-3.84 + 3.84i)T - 23iT^{2} \)
31 \( 1 + (-0.499 - 0.499i)T + 31iT^{2} \)
37 \( 1 + 6.77T + 37T^{2} \)
41 \( 1 + (3.62 - 3.62i)T - 41iT^{2} \)
43 \( 1 + 6.41T + 43T^{2} \)
47 \( 1 - 9.69T + 47T^{2} \)
53 \( 1 + (5.52 - 5.52i)T - 53iT^{2} \)
59 \( 1 - 10.5iT - 59T^{2} \)
61 \( 1 + (-7.92 - 7.92i)T + 61iT^{2} \)
67 \( 1 + (-0.259 + 0.259i)T - 67iT^{2} \)
71 \( 1 - 12.3iT - 71T^{2} \)
73 \( 1 - 2.56iT - 73T^{2} \)
79 \( 1 + (-0.678 + 0.678i)T - 79iT^{2} \)
83 \( 1 + (3.66 - 3.66i)T - 83iT^{2} \)
89 \( 1 + (12.5 - 12.5i)T - 89iT^{2} \)
97 \( 1 - 11.3T + 97T^{2} \)
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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

−9.500295471926108609607207632201, −8.811925580550820383149261840449, −8.435734146555816101822636597082, −7.07710356029122848011388108250, −6.74428558223003630493934095804, −5.62437623564862630688648949239, −4.98973124385352172719791661164, −4.15959569037399029970074127767, −2.75922975530498253079318892886, −1.40710978702583977873250786229, 0.66051560061058800049222990064, 1.71362358735541084137948490249, 3.31838498919530608998216713540, 3.81192452570428506810649952537, 5.00988788581311604932252560821, 5.84974302428385945399955619786, 6.62798097389525573297657076076, 7.934474513509599141600457347598, 8.503484085126884983772176471392, 9.168468868368552380309517864195

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