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Properties

Label 2-315-35.34-c4-0-15
Degree $2$
Conductor $315$
Sign $0.750 - 0.660i$
Analytic cond. $32.5615$
Root an. cond. $5.70627$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.52i·2-s − 14.4·4-s + (16.5 + 18.7i)5-s + (−0.191 + 48.9i)7-s − 8.41i·8-s + (103. − 91.5i)10-s + 186.·11-s − 295.·13-s + (270. + 1.05i)14-s − 278.·16-s − 330.·17-s + 241. i·19-s + (−240. − 270. i)20-s − 1.02e3i·22-s + 278. i·23-s + ⋯
L(s)  = 1  − 1.38i·2-s − 0.904·4-s + (0.663 + 0.748i)5-s + (−0.00389 + 0.999i)7-s − 0.131i·8-s + (1.03 − 0.915i)10-s + 1.54·11-s − 1.74·13-s + (1.38 + 0.00538i)14-s − 1.08·16-s − 1.14·17-s + 0.668i·19-s + (−0.600 − 0.676i)20-s − 2.12i·22-s + 0.526i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 - 0.660i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.750 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.750 - 0.660i$
Analytic conductor: \(32.5615\)
Root analytic conductor: \(5.70627\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :2),\ 0.750 - 0.660i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.327756388\)
\(L(\frac12)\) \(\approx\) \(1.327756388\)
\(L(3)\) 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 \( 1 + (-16.5 - 18.7i)T \)
7 \( 1 + (0.191 - 48.9i)T \)
good2 \( 1 + 5.52iT - 16T^{2} \)
11 \( 1 - 186.T + 1.46e4T^{2} \)
13 \( 1 + 295.T + 2.85e4T^{2} \)
17 \( 1 + 330.T + 8.35e4T^{2} \)
19 \( 1 - 241. iT - 1.30e5T^{2} \)
23 \( 1 - 278. iT - 2.79e5T^{2} \)
29 \( 1 + 407.T + 7.07e5T^{2} \)
31 \( 1 - 572. iT - 9.23e5T^{2} \)
37 \( 1 - 1.15e3iT - 1.87e6T^{2} \)
41 \( 1 - 1.30e3iT - 2.82e6T^{2} \)
43 \( 1 + 777. iT - 3.41e6T^{2} \)
47 \( 1 + 1.69e3T + 4.87e6T^{2} \)
53 \( 1 + 5.30e3iT - 7.89e6T^{2} \)
59 \( 1 - 2.77e3iT - 1.21e7T^{2} \)
61 \( 1 - 1.95e3iT - 1.38e7T^{2} \)
67 \( 1 - 5.89e3iT - 2.01e7T^{2} \)
71 \( 1 + 537.T + 2.54e7T^{2} \)
73 \( 1 + 6.80e3T + 2.83e7T^{2} \)
79 \( 1 - 1.11e4T + 3.89e7T^{2} \)
83 \( 1 - 9.75e3T + 4.74e7T^{2} \)
89 \( 1 + 5.80e3iT - 6.27e7T^{2} \)
97 \( 1 + 3.46e3T + 8.85e7T^{2} \)
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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

−11.38909170133365570193442931795, −10.12870190367066208974101243080, −9.601783648105943409840272694304, −8.833064050776303120278335727138, −7.09179544914243052904989072630, −6.26411052130139145400546755183, −4.85837074258902858916147927749, −3.49273648034315274466183925628, −2.42153987688843493134057458063, −1.65844959327794406687062291190, 0.37128755719481706196530856986, 2.08650272515085371892320277818, 4.29866880106337935699445306922, 4.94158285022239084451820904577, 6.26289675592862379539235765385, 6.92758530411901688377725852618, 7.79401170521259535241402066991, 9.072151563382090599756867402413, 9.463699663140666982636269032575, 10.82957310305534050865296028752

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