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Properties

Label 2-760-8.5-c1-0-25
Degree $2$
Conductor $760$
Sign $0.997 - 0.0752i$
Analytic cond. $6.06863$
Root an. cond. $2.46345$
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  + (−0.676 + 1.24i)2-s − 1.70i·3-s + (−1.08 − 1.67i)4-s − i·5-s + (2.11 + 1.15i)6-s + 1.91·7-s + (2.82 − 0.212i)8-s + 0.104·9-s + (1.24 + 0.676i)10-s + 4.53i·11-s + (−2.85 + 1.84i)12-s + 3.96i·13-s + (−1.29 + 2.38i)14-s − 1.70·15-s + (−1.64 + 3.64i)16-s + 1.87·17-s + ⋯
L(s)  = 1  + (−0.478 + 0.878i)2-s − 0.982i·3-s + (−0.542 − 0.839i)4-s − 0.447i·5-s + (0.862 + 0.469i)6-s + 0.724·7-s + (0.997 − 0.0752i)8-s + 0.0348·9-s + (0.392 + 0.213i)10-s + 1.36i·11-s + (−0.825 + 0.533i)12-s + 1.10i·13-s + (−0.346 + 0.636i)14-s − 0.439·15-s + (−0.410 + 0.911i)16-s + 0.454·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $0.997 - 0.0752i$
Analytic conductor: \(6.06863\)
Root analytic conductor: \(2.46345\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 760,\ (\ :1/2),\ 0.997 - 0.0752i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32600 + 0.0499350i\)
\(L(\frac12)\) \(\approx\) \(1.32600 + 0.0499350i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.676 - 1.24i)T \)
5 \( 1 + iT \)
19 \( 1 + iT \)
good3 \( 1 + 1.70iT - 3T^{2} \)
7 \( 1 - 1.91T + 7T^{2} \)
11 \( 1 - 4.53iT - 11T^{2} \)
13 \( 1 - 3.96iT - 13T^{2} \)
17 \( 1 - 1.87T + 17T^{2} \)
23 \( 1 - 5.88T + 23T^{2} \)
29 \( 1 + 9.11iT - 29T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 - 4.77iT - 37T^{2} \)
41 \( 1 - 1.03T + 41T^{2} \)
43 \( 1 - 6.88iT - 43T^{2} \)
47 \( 1 + 8.52T + 47T^{2} \)
53 \( 1 + 4.21iT - 53T^{2} \)
59 \( 1 + 5.57iT - 59T^{2} \)
61 \( 1 + 1.67iT - 61T^{2} \)
67 \( 1 + 8.44iT - 67T^{2} \)
71 \( 1 - 13.9T + 71T^{2} \)
73 \( 1 + 7.89T + 73T^{2} \)
79 \( 1 - 6.65T + 79T^{2} \)
83 \( 1 - 8.65iT - 83T^{2} \)
89 \( 1 + 4.77T + 89T^{2} \)
97 \( 1 + 3.50T + 97T^{2} \)
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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

−9.893988609551721675209627617173, −9.507625879926170673584227656110, −8.231797432074873260824077373263, −7.84353970680486682514898015525, −6.86255446743692611456681358656, −6.38397891276095312323927512643, −4.88409310523455659255547230682, −4.47962353864832269409528771288, −2.06605282190983943160543551931, −1.14283272180485575446405310001, 1.11715887439014112621510139210, 2.93985423284514253398154273255, 3.53016684690487222075131785832, 4.72654657046455120923912863339, 5.53958790929752704784458459656, 7.09227045879367523670219534629, 8.137161008645960112380038341480, 8.724363385000619986409134410433, 9.680940592665310556377917605028, 10.60569427086020552146496775895

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