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Properties

Label 2-755-755.603-c1-0-14
Degree $2$
Conductor $755$
Sign $-0.953 + 0.302i$
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  + (−0.812 + 0.812i)2-s + 0.678i·4-s + (−1.45 + 1.69i)5-s + (−2.17 − 2.17i)8-s + 3i·9-s + (−0.193 − 2.56i)10-s + 5.97·11-s + 2.18·16-s + (−4.13 + 4.13i)17-s + (−2.43 − 2.43i)18-s + 7.85i·19-s + (−1.15 − 0.988i)20-s + (−4.85 + 4.85i)22-s + (−0.752 − 4.94i)25-s − 9.05i·29-s + ⋯
L(s)  = 1  + (−0.574 + 0.574i)2-s + 0.339i·4-s + (−0.651 + 0.758i)5-s + (−0.769 − 0.769i)8-s + i·9-s + (−0.0613 − 0.810i)10-s + 1.80·11-s + 0.545·16-s + (−1.00 + 1.00i)17-s + (−0.574 − 0.574i)18-s + 1.80i·19-s + (−0.257 − 0.220i)20-s + (−1.03 + 1.03i)22-s + (−0.150 − 0.988i)25-s − 1.68i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.302i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 + 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(755\)    =    \(5 \cdot 151\)
Sign: $-0.953 + 0.302i$
Analytic conductor: \(6.02870\)
Root analytic conductor: \(2.45534\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{755} (603, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 755,\ (\ :1/2),\ -0.953 + 0.302i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.104753 - 0.676316i\)
\(L(\frac12)\) \(\approx\) \(0.104753 - 0.676316i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.45 - 1.69i)T \)
151 \( 1 - 12.2T \)
good2 \( 1 + (0.812 - 0.812i)T - 2iT^{2} \)
3 \( 1 - 3iT^{2} \)
7 \( 1 + 7iT^{2} \)
11 \( 1 - 5.97T + 11T^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + (4.13 - 4.13i)T - 17iT^{2} \)
19 \( 1 - 7.85iT - 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 + 9.05iT - 29T^{2} \)
31 \( 1 + 7.16T + 31T^{2} \)
37 \( 1 + (7.18 - 7.18i)T - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (5.63 + 5.63i)T + 43iT^{2} \)
47 \( 1 + (-9.34 + 9.34i)T - 47iT^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 - 15.3iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (8.92 - 8.92i)T - 97iT^{2} \)
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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

−10.64624939656253600941359234309, −9.960013947845864516866287062209, −8.743544229079496181258249888556, −8.289504544909974767954602590203, −7.36730391435726491946696266064, −6.69695479013271167755721547970, −5.88578341142018485415089853034, −4.06604054018839442659888761715, −3.67511355143679677279243546292, −1.98151785942284182315947121237, 0.43949426597181407238469611549, 1.52593456090209033830592456730, 3.16203782927367905311226056123, 4.29257104358163151961688679885, 5.24905627188714366085155586128, 6.54234134166452177932139497946, 7.16726949370704804095328199081, 8.836939559692111859928685491440, 9.069284123380376034148127870623, 9.456017261193764749280088491939

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