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Properties

Label 2-315-7.4-c1-0-2
Degree $2$
Conductor $315$
Sign $-0.993 - 0.112i$
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  + (1.32 + 2.29i)2-s + (−2.52 + 4.37i)4-s + (0.5 + 0.866i)5-s + (−2.35 + 1.20i)7-s − 8.10·8-s + (−1.32 + 2.29i)10-s + (2.19 − 3.80i)11-s + 3.39·13-s + (−5.90 − 3.80i)14-s + (−5.70 − 9.88i)16-s + (−1.32 + 2.29i)17-s + (3.68 + 6.37i)19-s − 5.05·20-s + 11.6·22-s + (−2.37 − 4.12i)23-s + ⋯
L(s)  = 1  + (0.938 + 1.62i)2-s + (−1.26 + 2.18i)4-s + (0.223 + 0.387i)5-s + (−0.889 + 0.456i)7-s − 2.86·8-s + (−0.419 + 0.727i)10-s + (0.662 − 1.14i)11-s + 0.941·13-s + (−1.57 − 1.01i)14-s + (−1.42 − 2.47i)16-s + (−0.322 + 0.557i)17-s + (0.844 + 1.46i)19-s − 1.12·20-s + 2.48·22-s + (−0.496 − 0.859i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.993 - 0.112i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.993 - 0.112i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.102920 + 1.82877i\)
\(L(\frac12)\) \(\approx\) \(0.102920 + 1.82877i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (2.35 - 1.20i)T \)
good2 \( 1 + (-1.32 - 2.29i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (-2.19 + 3.80i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.39T + 13T^{2} \)
17 \( 1 + (1.32 - 2.29i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.68 - 6.37i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.37 + 4.12i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.70T + 29T^{2} \)
31 \( 1 + (0.759 - 1.31i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.223 - 0.387i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.49T + 41T^{2} \)
43 \( 1 + 1.13T + 43T^{2} \)
47 \( 1 + (1.25 + 2.18i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.65 + 4.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.30 + 12.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.12 - 7.15i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.95 + 6.85i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.2T + 71T^{2} \)
73 \( 1 + (7.53 - 13.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.28 + 2.22i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.552T + 83T^{2} \)
89 \( 1 + (6.45 + 11.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.53T + 97T^{2} \)
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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

−12.50270226943557384005130744062, −11.51570896714280516471961787124, −10.04630284664599118162738626893, −8.776285498841957277693910787546, −8.236483045612908990622833654022, −6.87881231143537649571763388450, −6.10874810953832089903396636048, −5.69792338192610628999762320599, −3.99498216538527032029781735514, −3.23014103741360722219168224323, 1.11828419229209517751424452134, 2.64666726610654559022465758022, 3.85313628129206374508568373726, 4.69552996851458595905649911994, 5.89128405879752038034787870371, 7.07142531035242709320812885713, 9.108970403006991275767068430609, 9.556079810204737484070231998413, 10.43419269058973233059127095804, 11.45151117098413914420403398529

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