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Properties

Label 2-315-5.2-c2-0-21
Degree $2$
Conductor $315$
Sign $0.921 + 0.388i$
Analytic cond. $8.58312$
Root an. cond. $2.92969$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.867 + 0.867i)2-s − 2.49i·4-s + (4.93 + 0.833i)5-s + (−1.87 − 1.87i)7-s + (5.63 − 5.63i)8-s + (3.55 + 5.00i)10-s + 1.49·11-s + (2.15 − 2.15i)13-s − 3.24i·14-s − 0.198·16-s + (2.96 + 2.96i)17-s − 34.8i·19-s + (2.07 − 12.2i)20-s + (1.30 + 1.30i)22-s + (7.50 − 7.50i)23-s + ⋯
L(s)  = 1  + (0.433 + 0.433i)2-s − 0.623i·4-s + (0.986 + 0.166i)5-s + (−0.267 − 0.267i)7-s + (0.704 − 0.704i)8-s + (0.355 + 0.500i)10-s + 0.136·11-s + (0.165 − 0.165i)13-s − 0.231i·14-s − 0.0124·16-s + (0.174 + 0.174i)17-s − 1.83i·19-s + (0.103 − 0.614i)20-s + (0.0591 + 0.0591i)22-s + (0.326 − 0.326i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.388i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.921 + 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.921 + 0.388i$
Analytic conductor: \(8.58312\)
Root analytic conductor: \(2.92969\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1),\ 0.921 + 0.388i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.37981 - 0.481546i\)
\(L(\frac12)\) \(\approx\) \(2.37981 - 0.481546i\)
\(L(2)\) 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 \( 1 + (-4.93 - 0.833i)T \)
7 \( 1 + (1.87 + 1.87i)T \)
good2 \( 1 + (-0.867 - 0.867i)T + 4iT^{2} \)
11 \( 1 - 1.49T + 121T^{2} \)
13 \( 1 + (-2.15 + 2.15i)T - 169iT^{2} \)
17 \( 1 + (-2.96 - 2.96i)T + 289iT^{2} \)
19 \( 1 + 34.8iT - 361T^{2} \)
23 \( 1 + (-7.50 + 7.50i)T - 529iT^{2} \)
29 \( 1 - 37.1iT - 841T^{2} \)
31 \( 1 - 47.0T + 961T^{2} \)
37 \( 1 + (-16.3 - 16.3i)T + 1.36e3iT^{2} \)
41 \( 1 + 73.4T + 1.68e3T^{2} \)
43 \( 1 + (0.244 - 0.244i)T - 1.84e3iT^{2} \)
47 \( 1 + (-38.9 - 38.9i)T + 2.20e3iT^{2} \)
53 \( 1 + (-33.0 + 33.0i)T - 2.80e3iT^{2} \)
59 \( 1 - 31.6iT - 3.48e3T^{2} \)
61 \( 1 + 106.T + 3.72e3T^{2} \)
67 \( 1 + (-28.6 - 28.6i)T + 4.48e3iT^{2} \)
71 \( 1 + 15.8T + 5.04e3T^{2} \)
73 \( 1 + (26.2 - 26.2i)T - 5.32e3iT^{2} \)
79 \( 1 - 73.8iT - 6.24e3T^{2} \)
83 \( 1 + (58.6 - 58.6i)T - 6.88e3iT^{2} \)
89 \( 1 - 83.2iT - 7.92e3T^{2} \)
97 \( 1 + (103. + 103. i)T + 9.40e3iT^{2} \)
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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

−11.15802191965202635524265114106, −10.39158611037957245829067734535, −9.634553474276218063317838699221, −8.686669691087679630633243403980, −7.02427317815190282897726218461, −6.51503612818603636331894829116, −5.43614290336672299767171298952, −4.55998216687888192959300160620, −2.83506882939301841988948249038, −1.15815067450848809865654822691, 1.76797417717134403641842373046, 2.99779953624930446839538698571, 4.21931235914367796894753285356, 5.47155979568818500978133865033, 6.43244615517479526023769217018, 7.78183628831066145527768611273, 8.696833503850189551320898297338, 9.769485707797908110556311748673, 10.54819201855282469084252506950, 11.86550045889955197174210739133

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