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Properties

Label 2-715-13.10-c1-0-19
Degree $2$
Conductor $715$
Sign $0.921 - 0.389i$
Analytic cond. $5.70930$
Root an. cond. $2.38941$
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  + (0.809 − 0.467i)2-s + (0.346 + 0.600i)3-s + (−0.563 + 0.975i)4-s − i·5-s + (0.561 + 0.323i)6-s + (0.420 + 0.242i)7-s + 2.92i·8-s + (1.25 − 2.18i)9-s + (−0.467 − 0.809i)10-s + (0.866 − 0.5i)11-s − 0.781·12-s + (3.58 + 0.398i)13-s + 0.453·14-s + (0.600 − 0.346i)15-s + (0.238 + 0.412i)16-s + (−2.35 + 4.08i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.330i)2-s + (0.200 + 0.346i)3-s + (−0.281 + 0.487i)4-s − 0.447i·5-s + (0.229 + 0.132i)6-s + (0.158 + 0.0916i)7-s + 1.03i·8-s + (0.419 − 0.727i)9-s + (−0.147 − 0.255i)10-s + (0.261 − 0.150i)11-s − 0.225·12-s + (0.993 + 0.110i)13-s + 0.121·14-s + (0.155 − 0.0895i)15-s + (0.0596 + 0.103i)16-s + (−0.571 + 0.990i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 715 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.389i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 715 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 - 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(715\)    =    \(5 \cdot 11 \cdot 13\)
Sign: $0.921 - 0.389i$
Analytic conductor: \(5.70930\)
Root analytic conductor: \(2.38941\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{715} (166, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 715,\ (\ :1/2),\ 0.921 - 0.389i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.14620 + 0.435175i\)
\(L(\frac12)\) \(\approx\) \(2.14620 + 0.435175i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + iT \)
11 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (-3.58 - 0.398i)T \)
good2 \( 1 + (-0.809 + 0.467i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.346 - 0.600i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-0.420 - 0.242i)T + (3.5 + 6.06i)T^{2} \)
17 \( 1 + (2.35 - 4.08i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.42 - 3.13i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.823 + 1.42i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.12 - 8.88i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.89iT - 31T^{2} \)
37 \( 1 + (1.49 - 0.860i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.44 - 1.40i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.07 + 1.85i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.90iT - 47T^{2} \)
53 \( 1 - 9.25T + 53T^{2} \)
59 \( 1 + (8.67 + 5.00i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.15 + 12.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.20 - 3.57i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.12 - 2.38i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 8.36iT - 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 + 0.355iT - 83T^{2} \)
89 \( 1 + (0.495 - 0.285i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.73 + 3.31i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.56501562062997627538715485274, −9.528315127536529714300498542233, −8.663082073788742590177272401103, −8.239228488804142522609707236234, −6.89338979932482520577861223129, −5.81242599626852420527185674469, −4.75989426043177653303793027802, −3.86933503346571405333193296231, −3.24446442248962788636160326208, −1.50762941801019155513996975477, 1.15710902805787343279257548332, 2.70517776343320022093003994058, 4.06130999445226389896646016540, 4.92210568918920060975472118907, 5.88792856116413756377794428750, 6.85821107849184220762846478490, 7.46709501021944438600141740876, 8.615579068041926335797341344923, 9.583133350060713247902096749990, 10.32477819028496157403037931716

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