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Properties

Label 2-315-35.33-c1-0-13
Degree $2$
Conductor $315$
Sign $0.781 + 0.624i$
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  + (0.173 + 0.648i)2-s + (1.34 − 0.774i)4-s + (−1.64 − 1.51i)5-s + (−0.588 − 2.57i)7-s + (1.68 + 1.68i)8-s + (0.692 − 1.33i)10-s + (−0.0701 − 0.121i)11-s + (2.35 − 2.35i)13-s + (1.56 − 0.829i)14-s + (0.750 − 1.29i)16-s + (1.97 − 7.37i)17-s + (−3.89 + 6.74i)19-s + (−3.38 − 0.750i)20-s + (0.0665 − 0.0665i)22-s + (2.50 − 0.671i)23-s + ⋯
L(s)  = 1  + (0.122 + 0.458i)2-s + (0.670 − 0.387i)4-s + (−0.737 − 0.675i)5-s + (−0.222 − 0.974i)7-s + (0.595 + 0.595i)8-s + (0.219 − 0.420i)10-s + (−0.0211 − 0.0366i)11-s + (0.653 − 0.653i)13-s + (0.419 − 0.221i)14-s + (0.187 − 0.324i)16-s + (0.478 − 1.78i)17-s + (−0.893 + 1.54i)19-s + (−0.756 − 0.167i)20-s + (0.0141 − 0.0141i)22-s + (0.522 − 0.140i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 + 0.624i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.781 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.781 + 0.624i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.781 + 0.624i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36697 - 0.478995i\)
\(L(\frac12)\) \(\approx\) \(1.36697 - 0.478995i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.64 + 1.51i)T \)
7 \( 1 + (0.588 + 2.57i)T \)
good2 \( 1 + (-0.173 - 0.648i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (0.0701 + 0.121i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.35 + 2.35i)T - 13iT^{2} \)
17 \( 1 + (-1.97 + 7.37i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.89 - 6.74i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.50 + 0.671i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 5.09iT - 29T^{2} \)
31 \( 1 + (-2.54 + 1.46i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.53 - 5.73i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 0.261iT - 41T^{2} \)
43 \( 1 + (2.11 + 2.11i)T + 43iT^{2} \)
47 \( 1 + (1.50 - 0.402i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.749 - 2.79i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-4.37 - 7.57i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.76 + 2.75i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.54 - 2.02i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 3.56T + 71T^{2} \)
73 \( 1 + (3.16 + 0.847i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-0.113 - 0.0656i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.33 - 7.33i)T - 83iT^{2} \)
89 \( 1 + (2.44 - 4.23i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.25 + 1.25i)T + 97iT^{2} \)
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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

−11.51667441105559136895989577995, −10.68527052197592648258673171745, −9.839027707238482676378571864628, −8.424217279717366479656722661804, −7.63157771331574644419859989377, −6.82107159521186417252872991473, −5.62152529206082727504653951882, −4.56882073937838721571466643973, −3.24471140777952431518082983341, −1.10595271858994796463878154655, 2.13453849154106985289927014578, 3.25921556447759510630229074391, 4.29157894316005491487445720966, 6.13791788455672261972244897303, 6.79923430969525364511279880019, 8.001642187918073351313359885120, 8.820107735778822005725879382337, 10.20824334730118193933478234033, 11.12803017407264145179712663245, 11.56737099258160187880903210656

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