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Properties

Label 2-315-35.12-c1-0-4
Degree $2$
Conductor $315$
Sign $0.428 - 0.903i$
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  + (−1.72 + 0.461i)2-s + (1.01 − 0.587i)4-s + (1.95 + 1.09i)5-s + (1.68 + 2.04i)7-s + (1.03 − 1.03i)8-s + (−3.86 − 0.983i)10-s + (−1.46 − 2.54i)11-s + (−0.187 − 0.187i)13-s + (−3.83 − 2.74i)14-s + (−2.48 + 4.30i)16-s + (3.24 + 0.868i)17-s + (1.81 − 3.14i)19-s + (2.62 − 0.0328i)20-s + (3.69 + 3.69i)22-s + (2.43 + 9.07i)23-s + ⋯
L(s)  = 1  + (−1.21 + 0.326i)2-s + (0.508 − 0.293i)4-s + (0.872 + 0.489i)5-s + (0.635 + 0.772i)7-s + (0.367 − 0.367i)8-s + (−1.22 − 0.310i)10-s + (−0.442 − 0.766i)11-s + (−0.0521 − 0.0521i)13-s + (−1.02 − 0.732i)14-s + (−0.621 + 1.07i)16-s + (0.786 + 0.210i)17-s + (0.417 − 0.722i)19-s + (0.587 − 0.00734i)20-s + (0.788 + 0.788i)22-s + (0.507 + 1.89i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.428 - 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.428 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.428 - 0.903i$
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: \((2,\ 315,\ (\ :1/2),\ 0.428 - 0.903i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.709541 + 0.448649i\)
\(L(\frac12)\) \(\approx\) \(0.709541 + 0.448649i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.95 - 1.09i)T \)
7 \( 1 + (-1.68 - 2.04i)T \)
good2 \( 1 + (1.72 - 0.461i)T + (1.73 - i)T^{2} \)
11 \( 1 + (1.46 + 2.54i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.187 + 0.187i)T + 13iT^{2} \)
17 \( 1 + (-3.24 - 0.868i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.81 + 3.14i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.43 - 9.07i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 0.815iT - 29T^{2} \)
31 \( 1 + (3.76 - 2.17i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.31 + 1.69i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 2.45iT - 41T^{2} \)
43 \( 1 + (3.59 - 3.59i)T - 43iT^{2} \)
47 \( 1 + (-2.47 - 9.24i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (4.87 + 1.30i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (5.41 + 9.37i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.07 - 4.66i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.10 + 15.3i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 2.68T + 71T^{2} \)
73 \( 1 + (-0.508 + 1.89i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (4.44 + 2.56i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.09 + 6.09i)T + 83iT^{2} \)
89 \( 1 + (-4.87 + 8.43i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.93 - 5.93i)T - 97iT^{2} \)
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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

−11.35794873691391563560467034125, −10.79152500927377604095004084731, −9.616829416532023789746787203343, −9.174071340756091439970797651263, −8.078672569348423621802512468736, −7.32834700399471178533432317201, −6.04936270795707950169771823586, −5.16346720921216002257906801670, −3.12345428439067141402857469277, −1.53433427013460576984761720872, 1.06399575871526783013452273653, 2.31050487755843213918125051400, 4.47591253619715443313966235798, 5.42223696037213276471167753431, 7.00061708817089320158299168806, 7.958533026635280376121317890586, 8.728105790880652919695738375108, 9.905308460709258037516778003537, 10.15293439465439405334347435639, 11.14019791693251944700331614714

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