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Properties

Label 2-755-5.4-c1-0-62
Degree $2$
Conductor $755$
Sign $0.105 + 0.994i$
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  − 0.250i·2-s + 0.442i·3-s + 1.93·4-s + (−0.235 − 2.22i)5-s + 0.110·6-s − 1.63i·7-s − 0.985i·8-s + 2.80·9-s + (−0.556 + 0.0588i)10-s − 5.60·11-s + 0.858i·12-s − 1.36i·13-s − 0.410·14-s + (0.984 − 0.104i)15-s + 3.62·16-s − 6.74i·17-s + ⋯
L(s)  = 1  − 0.176i·2-s + 0.255i·3-s + 0.968·4-s + (−0.105 − 0.994i)5-s + 0.0452·6-s − 0.619i·7-s − 0.348i·8-s + 0.934·9-s + (−0.175 + 0.0186i)10-s − 1.69·11-s + 0.247i·12-s − 0.377i·13-s − 0.109·14-s + (0.254 − 0.0269i)15-s + 0.907·16-s − 1.63i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(755\)    =    \(5 \cdot 151\)
Sign: $0.105 + 0.994i$
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: \((2,\ 755,\ (\ :1/2),\ 0.105 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30373 - 1.17307i\)
\(L(\frac12)\) \(\approx\) \(1.30373 - 1.17307i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.235 + 2.22i)T \)
151 \( 1 + T \)
good2 \( 1 + 0.250iT - 2T^{2} \)
3 \( 1 - 0.442iT - 3T^{2} \)
7 \( 1 + 1.63iT - 7T^{2} \)
11 \( 1 + 5.60T + 11T^{2} \)
13 \( 1 + 1.36iT - 13T^{2} \)
17 \( 1 + 6.74iT - 17T^{2} \)
19 \( 1 + 0.591T + 19T^{2} \)
23 \( 1 - 2.58iT - 23T^{2} \)
29 \( 1 - 6.70T + 29T^{2} \)
31 \( 1 + 7.88T + 31T^{2} \)
37 \( 1 - 3.52iT - 37T^{2} \)
41 \( 1 - 3.19T + 41T^{2} \)
43 \( 1 - 11.8iT - 43T^{2} \)
47 \( 1 + 7.93iT - 47T^{2} \)
53 \( 1 + 11.3iT - 53T^{2} \)
59 \( 1 - 4.11T + 59T^{2} \)
61 \( 1 - 8.31T + 61T^{2} \)
67 \( 1 + 9.36iT - 67T^{2} \)
71 \( 1 - 11.6T + 71T^{2} \)
73 \( 1 - 11.1iT - 73T^{2} \)
79 \( 1 - 5.19T + 79T^{2} \)
83 \( 1 - 3.94iT - 83T^{2} \)
89 \( 1 + 17.8T + 89T^{2} \)
97 \( 1 + 5.09iT - 97T^{2} \)
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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

−10.08981509283661374327200975661, −9.641738322849345904766837406760, −8.238522800705814154473124737035, −7.55478654190111491512224844364, −6.87230556949962694614744806653, −5.41747964002810975107737488506, −4.82505010637232825577023002590, −3.55754449868791099597288543473, −2.34643231180975377318999023376, −0.877519970642938233558487678818, 1.98121494225564122066304245614, 2.67607339406946285849369037504, 3.98093292909405736546090966146, 5.50452271503474316034319434819, 6.25019920753567665547816924157, 7.12197151674008213981217392125, 7.72019300868299310392055525197, 8.582676636538905531110823419234, 10.07939603682858956974717668972, 10.59354254344767668451852103257

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