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Properties

Label 1-1520-1520.797-r0-0-0
Degree $1$
Conductor $1520$
Sign $0.584 + 0.811i$
Analytic cond. $7.05885$
Root an. cond. $7.05885$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − i·7-s + 9-s − i·11-s − 13-s + i·17-s + i·21-s + i·23-s − 27-s + i·29-s − 31-s + i·33-s − 37-s + 39-s + 41-s + ⋯
L(s)  = 1  − 3-s − i·7-s + 9-s − i·11-s − 13-s + i·17-s + i·21-s + i·23-s − 27-s + i·29-s − 31-s + i·33-s − 37-s + 39-s + 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $0.584 + 0.811i$
Analytic conductor: \(7.05885\)
Root analytic conductor: \(7.05885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (797, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1520,\ (0:\ ),\ 0.584 + 0.811i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6024409513 + 0.3084005439i\)
\(L(\frac12)\) \(\approx\) \(0.6024409513 + 0.3084005439i\)
\(L(1)\) \(\approx\) \(0.6938157994 + 0.02109565150i\)
\(L(1)\) \(\approx\) \(0.6938157994 + 0.02109565150i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good3 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 \)
13 \( 1 \)
17 \( 1 \)
23 \( 1 \)
29 \( 1 + T \)
31 \( 1 \)
37 \( 1 - iT \)
41 \( 1 \)
43 \( 1 - T \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 + iT \)
67 \( 1 \)
71 \( 1 \)
73 \( 1 \)
79 \( 1 + iT \)
83 \( 1 \)
89 \( 1 + iT \)
97 \( 1 \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.58869468705282910665586293981, −19.6919729300045294169139037368, −18.73153949831517251972270830106, −18.20811138523069456010280549295, −17.55501160296969128206045977656, −16.766621720713631841984199527, −16.05951429573127373544616554792, −15.216846154957576761936041158433, −14.73174032044551531040474281856, −13.54089257186000850063135870669, −12.5157682702270701789529267875, −12.18332815624751871511280294847, −11.51967608829204570418780893939, −10.529323430990370977915178152184, −9.74303342359389698695821141362, −9.16384465740343745117848750204, −7.930848708141140778650184682157, −7.07262841482462241177921185695, −6.44672602517454718077972084898, −5.289286400385436623698440955668, −5.00542771686988311546529923604, −3.97692261629954981423499996422, −2.54351578639429131445716179112, −1.90636004248462474032241749801, −0.363031836805342870910948049494, 0.89201192489592899446285257073, 1.83182155997618807449697148962, 3.36259524218234770431935947478, 4.04587769032138997277762266519, 5.077981212739702399357603714218, 5.71188878113241838847335797103, 6.70542989214947556148182272279, 7.31355723435015115867617097498, 8.1792923221635765685126459955, 9.351013970773898594533839877768, 10.17424339854017090414419965367, 10.85405852158762050522498398276, 11.37686701851880820516813325870, 12.4041718358032512831712148266, 12.98836541600753503441570337263, 13.86319194527726280601297690740, 14.64953524938740208402296323059, 15.63373503129069265604736986459, 16.44909474789039258311688891617, 16.955903936711730695231296041390, 17.53864956381005139118777399268, 18.338081726674871695379203471424, 19.346086480284870208790186158, 19.72809335742125530418059530152, 20.86863554619500382248534786028

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