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Properties

Label 2-755-151.64-c1-0-29
Degree $2$
Conductor $755$
Sign $0.154 - 0.988i$
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  + 2.53·2-s + (−1.99 + 1.44i)3-s + 4.42·4-s + (−0.309 + 0.951i)5-s + (−5.04 + 3.66i)6-s + (0.251 − 0.772i)7-s + 6.14·8-s + (0.943 − 2.90i)9-s + (−0.783 + 2.41i)10-s + (5.02 + 3.65i)11-s + (−8.80 + 6.39i)12-s + (−2.09 + 1.52i)13-s + (0.636 − 1.95i)14-s + (−0.760 − 2.33i)15-s + 6.71·16-s + (0.459 + 1.41i)17-s + ⋯
L(s)  = 1  + 1.79·2-s + (−1.14 + 0.834i)3-s + 2.21·4-s + (−0.138 + 0.425i)5-s + (−2.05 + 1.49i)6-s + (0.0949 − 0.292i)7-s + 2.17·8-s + (0.314 − 0.967i)9-s + (−0.247 + 0.762i)10-s + (1.51 + 1.10i)11-s + (−2.54 + 1.84i)12-s + (−0.581 + 0.422i)13-s + (0.170 − 0.523i)14-s + (−0.196 − 0.604i)15-s + 1.67·16-s + (0.111 + 0.343i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.154 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.154 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(755\)    =    \(5 \cdot 151\)
Sign: $0.154 - 0.988i$
Analytic conductor: \(6.02870\)
Root analytic conductor: \(2.45534\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{755} (366, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 755,\ (\ :1/2),\ 0.154 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.32213 + 1.98739i\)
\(L(\frac12)\) \(\approx\) \(2.32213 + 1.98739i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.309 - 0.951i)T \)
151 \( 1 + (-7.05 + 10.0i)T \)
good2 \( 1 - 2.53T + 2T^{2} \)
3 \( 1 + (1.99 - 1.44i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + (-0.251 + 0.772i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (-5.02 - 3.65i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (2.09 - 1.52i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.459 - 1.41i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + 4.80T + 19T^{2} \)
23 \( 1 - 1.20T + 23T^{2} \)
29 \( 1 + (-0.968 - 0.703i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.60 - 8.01i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-1.17 + 0.854i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-5.42 + 3.94i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (3.25 + 10.0i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-6.96 + 5.05i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.03 + 2.20i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 - 1.90T + 59T^{2} \)
61 \( 1 + (8.05 + 5.85i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-1.42 - 4.38i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-3.81 + 11.7i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.995 - 3.06i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-3.64 + 11.2i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.61 + 4.95i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (0.143 + 0.441i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (5.15 + 15.8i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.72057128456611575588621741903, −10.27463422584803141791567745747, −9.054064189543096760169525296513, −7.23375950037597650572488084717, −6.67532539075703249448832866062, −5.94387702535030806400948270164, −4.85035405488909191855607184184, −4.34814714182920681033010859844, −3.63500227658198291486494581501, −2.02876127425222561914302536732, 1.08440181399696018235229690561, 2.60144706699956304523556545304, 3.95899000041493313207305756269, 4.79709960582168540657081356751, 5.87040158978109114603826791881, 6.15650849050887512270992249128, 6.99358709832839848983722490555, 8.061465289283234746413969146151, 9.349000456260148859588167989760, 10.87850176611523169068175327582

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