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Properties

Label 2-315-35.12-c1-0-14
Degree $2$
Conductor $315$
Sign $0.872 + 0.489i$
Analytic cond. $2.51528$
Root an. cond. $1.58596$
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  + (2.24 − 0.602i)2-s + (2.95 − 1.70i)4-s + (2.22 + 0.198i)5-s + (−2.59 + 0.519i)7-s + (2.33 − 2.33i)8-s + (5.12 − 0.895i)10-s + (1.76 + 3.05i)11-s + (−4.49 − 4.49i)13-s + (−5.51 + 2.73i)14-s + (0.421 − 0.729i)16-s + (−1.79 − 0.481i)17-s + (0.0699 − 0.121i)19-s + (6.92 − 3.21i)20-s + (5.80 + 5.80i)22-s + (−0.997 − 3.72i)23-s + ⋯
L(s)  = 1  + (1.58 − 0.425i)2-s + (1.47 − 0.854i)4-s + (0.996 + 0.0886i)5-s + (−0.980 + 0.196i)7-s + (0.824 − 0.824i)8-s + (1.62 − 0.283i)10-s + (0.531 + 0.921i)11-s + (−1.24 − 1.24i)13-s + (−1.47 + 0.730i)14-s + (0.105 − 0.182i)16-s + (−0.436 − 0.116i)17-s + (0.0160 − 0.0277i)19-s + (1.54 − 0.719i)20-s + (1.23 + 1.23i)22-s + (−0.207 − 0.775i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.872 + 0.489i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.872 + 0.489i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.95029 - 0.771243i\)
\(L(\frac12)\) \(\approx\) \(2.95029 - 0.771243i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-2.22 - 0.198i)T \)
7 \( 1 + (2.59 - 0.519i)T \)
good2 \( 1 + (-2.24 + 0.602i)T + (1.73 - i)T^{2} \)
11 \( 1 + (-1.76 - 3.05i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.49 + 4.49i)T + 13iT^{2} \)
17 \( 1 + (1.79 + 0.481i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.0699 + 0.121i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.997 + 3.72i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 2.01iT - 29T^{2} \)
31 \( 1 + (4.56 - 2.63i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.61 + 1.50i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 0.903iT - 41T^{2} \)
43 \( 1 + (-2.38 + 2.38i)T - 43iT^{2} \)
47 \( 1 + (-0.639 - 2.38i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.71 + 0.726i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.15 - 5.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.69 + 5.01i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.77 - 10.3i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 5.09T + 71T^{2} \)
73 \( 1 + (-2.42 + 9.04i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (7.30 + 4.21i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.37 - 7.37i)T + 83iT^{2} \)
89 \( 1 + (-1.75 + 3.03i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.70 + 8.70i)T - 97iT^{2} \)
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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

−12.07192614810831675965159521320, −10.71751836191832820254531588988, −9.996449302908438901190909683377, −9.105282642173303705094116465993, −7.22518258256061467069801345962, −6.33510596851882040111649343484, −5.46713343436087511630482546727, −4.52251138251834472634524635373, −3.08997233333791489958209111146, −2.22339379330318996273747305987, 2.35873990694492528665098085872, 3.61584900517285356678928979228, 4.71259532464776571243623026373, 5.88798499756974232996467866308, 6.44551879671953099441275904770, 7.34647977533541609521612242904, 9.148285898172312884875940622706, 9.743814884519817706179244642674, 11.16663312968457575486733494569, 12.06502232071852428229401589801

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