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Properties

Label 2-315-9.7-c1-0-17
Degree $2$
Conductor $315$
Sign $0.710 + 0.703i$
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  + (0.627 − 1.08i)2-s + (1.73 + 0.0710i)3-s + (0.211 + 0.367i)4-s + (−0.5 − 0.866i)5-s + (1.16 − 1.83i)6-s + (0.5 − 0.866i)7-s + 3.04·8-s + (2.98 + 0.245i)9-s − 1.25·10-s + (−2.97 + 5.14i)11-s + (0.340 + 0.650i)12-s + (−1.86 − 3.22i)13-s + (−0.627 − 1.08i)14-s + (−0.803 − 1.53i)15-s + (1.48 − 2.57i)16-s − 6.19·17-s + ⋯
L(s)  = 1  + (0.443 − 0.768i)2-s + (0.999 + 0.0410i)3-s + (0.105 + 0.183i)4-s + (−0.223 − 0.387i)5-s + (0.475 − 0.749i)6-s + (0.188 − 0.327i)7-s + 1.07·8-s + (0.996 + 0.0819i)9-s − 0.397·10-s + (−0.895 + 1.55i)11-s + (0.0983 + 0.187i)12-s + (−0.516 − 0.894i)13-s + (−0.167 − 0.290i)14-s + (−0.207 − 0.396i)15-s + (0.371 − 0.643i)16-s − 1.50·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.710 + 0.703i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.710 + 0.703i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.10859 - 0.866962i\)
\(L(\frac12)\) \(\approx\) \(2.10859 - 0.866962i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.73 - 0.0710i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.627 + 1.08i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (2.97 - 5.14i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.86 + 3.22i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 6.19T + 17T^{2} \)
19 \( 1 - 5.61T + 19T^{2} \)
23 \( 1 + (2.98 + 5.16i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.86 - 3.22i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.11 + 1.93i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.36T + 37T^{2} \)
41 \( 1 + (0.0273 + 0.0472i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.71 - 4.69i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.73 + 3.00i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 1.74T + 53T^{2} \)
59 \( 1 + (-4.83 - 8.37i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.41 - 5.91i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.411 + 0.712i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.2T + 71T^{2} \)
73 \( 1 - 1.27T + 73T^{2} \)
79 \( 1 + (2.54 - 4.40i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.88 + 6.72i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 4.88T + 89T^{2} \)
97 \( 1 + (-3.74 + 6.49i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.73738638333014662037806787701, −10.48982328491173762107331441845, −9.961466394629186595992628732881, −8.685326732456004306838343937580, −7.61814827433321213259724954101, −7.21035614718369888817801394044, −4.91657980234482736266854620511, −4.26575067600326559622635315745, −2.92951599400670932751166921353, −1.95563545243913091910465272150, 2.09074875150293608343684261772, 3.47071182510842380989034503754, 4.80819110651174488797531619025, 5.93205099197375247339210009045, 7.04175296318468138171045964465, 7.78465364926801376433166739415, 8.739743049435950290430075605122, 9.783106603678047228584759499793, 10.89395651847454496227600310598, 11.67316768059699773695032341712

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