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Properties

Label 2-755-5.4-c1-0-58
Degree $2$
Conductor $755$
Sign $-0.964 - 0.265i$
Analytic cond. $6.02870$
Root an. cond. $2.45534$
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  − 2.54i·2-s − 0.456i·3-s − 4.49·4-s + (2.15 + 0.594i)5-s − 1.16·6-s − 3.74i·7-s + 6.35i·8-s + 2.79·9-s + (1.51 − 5.49i)10-s + 1.77·11-s + 2.05i·12-s + 0.0609i·13-s − 9.55·14-s + (0.271 − 0.984i)15-s + 7.19·16-s − 3.32i·17-s + ⋯
L(s)  = 1  − 1.80i·2-s − 0.263i·3-s − 2.24·4-s + (0.964 + 0.265i)5-s − 0.475·6-s − 1.41i·7-s + 2.24i·8-s + 0.930·9-s + (0.479 − 1.73i)10-s + 0.536·11-s + 0.592i·12-s + 0.0169i·13-s − 2.55·14-s + (0.0701 − 0.254i)15-s + 1.79·16-s − 0.806i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(755\)    =    \(5 \cdot 151\)
Sign: $-0.964 - 0.265i$
Analytic conductor: \(6.02870\)
Root analytic conductor: \(2.45534\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{755} (454, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 755,\ (\ :1/2),\ -0.964 - 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.218258 + 1.61214i\)
\(L(\frac12)\) \(\approx\) \(0.218258 + 1.61214i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.15 - 0.594i)T \)
151 \( 1 + T \)
good2 \( 1 + 2.54iT - 2T^{2} \)
3 \( 1 + 0.456iT - 3T^{2} \)
7 \( 1 + 3.74iT - 7T^{2} \)
11 \( 1 - 1.77T + 11T^{2} \)
13 \( 1 - 0.0609iT - 13T^{2} \)
17 \( 1 + 3.32iT - 17T^{2} \)
19 \( 1 + 2.26T + 19T^{2} \)
23 \( 1 + 4.84iT - 23T^{2} \)
29 \( 1 + 4.33T + 29T^{2} \)
31 \( 1 - 4.39T + 31T^{2} \)
37 \( 1 - 3.10iT - 37T^{2} \)
41 \( 1 + 8.84T + 41T^{2} \)
43 \( 1 - 9.65iT - 43T^{2} \)
47 \( 1 + 0.624iT - 47T^{2} \)
53 \( 1 - 7.24iT - 53T^{2} \)
59 \( 1 - 12.0T + 59T^{2} \)
61 \( 1 + 10.0T + 61T^{2} \)
67 \( 1 + 7.02iT - 67T^{2} \)
71 \( 1 - 2.29T + 71T^{2} \)
73 \( 1 - 1.91iT - 73T^{2} \)
79 \( 1 - 1.66T + 79T^{2} \)
83 \( 1 - 6.23iT - 83T^{2} \)
89 \( 1 + 15.6T + 89T^{2} \)
97 \( 1 - 1.85iT - 97T^{2} \)
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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

−9.980742022581481332599016677346, −9.644841292773629828441290322651, −8.531478942965231950035328823501, −7.23756750627434800069934204756, −6.43014060258806182523595555178, −4.80665297318100776668138852140, −4.15555201363890680110424079525, −3.03396799895201808806163495385, −1.83669460819413262055423885425, −0.920402854148045160263503995585, 1.87539276813057543598763635950, 3.85230554290193938915052824058, 4.99796295125429362787943515285, 5.63654975585813797205331305564, 6.33675572108501221379314216099, 7.13645732400108970806900365237, 8.332384170914111199411202858881, 8.920809503503706772734696658644, 9.541690897973243269045445423885, 10.31127170686411233843883087493

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