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Properties

Label 2-315-7.4-c1-0-9
Degree $2$
Conductor $315$
Sign $0.947 - 0.318i$
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  + (0.707 + 1.22i)2-s + (−0.5 − 0.866i)5-s + (1.62 − 2.09i)7-s + 2.82·8-s + (0.707 − 1.22i)10-s + (1.70 − 2.95i)11-s − 1.58·13-s + (3.70 + 0.507i)14-s + (2.00 + 3.46i)16-s + (−3.12 + 5.40i)17-s + (3.32 + 5.76i)19-s + 4.82·22-s + (−3.12 − 5.40i)23-s + (−0.499 + 0.866i)25-s + (−1.12 − 1.94i)26-s + ⋯
L(s)  = 1  + (0.499 + 0.866i)2-s + (−0.223 − 0.387i)5-s + (0.612 − 0.790i)7-s + 0.999·8-s + (0.223 − 0.387i)10-s + (0.514 − 0.891i)11-s − 0.439·13-s + (0.990 + 0.135i)14-s + (0.500 + 0.866i)16-s + (−0.757 + 1.31i)17-s + (0.763 + 1.32i)19-s + 1.02·22-s + (−0.650 − 1.12i)23-s + (−0.0999 + 0.173i)25-s + (−0.219 − 0.380i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.947 - 0.318i$
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: \((2,\ 315,\ (\ :1/2),\ 0.947 - 0.318i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84001 + 0.300926i\)
\(L(\frac12)\) \(\approx\) \(1.84001 + 0.300926i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 + 2.09i)T \)
good2 \( 1 + (-0.707 - 1.22i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (-1.70 + 2.95i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.58T + 13T^{2} \)
17 \( 1 + (3.12 - 5.40i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.32 - 5.76i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.12 + 5.40i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.242T + 29T^{2} \)
31 \( 1 + (-0.0857 + 0.148i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.79 - 4.83i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.24T + 41T^{2} \)
43 \( 1 + 10.4T + 43T^{2} \)
47 \( 1 + (-4.65 - 8.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.585 + 1.01i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.707 + 1.22i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.24 + 10.8i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.86 - 10.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.41T + 71T^{2} \)
73 \( 1 + (1.03 - 1.79i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.32 - 4.03i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.41T + 83T^{2} \)
89 \( 1 + (1.87 + 3.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.8T + 97T^{2} \)
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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

−11.71164066448348379248031098438, −10.77196273631920059902294916584, −9.994629222192400025889297957654, −8.416777543497311114334377257040, −7.891052026420932181237098485340, −6.69030570590213875083048281646, −5.87676115550160208194034638682, −4.69328974760513912708376762410, −3.84468998392254083656486381308, −1.48496743719243089836090257714, 1.98947451530844239465567255704, 2.98404141632036521417377373953, 4.39247059935920375892407219364, 5.22513359482241011609849043808, 6.96706377875793270950840113730, 7.59800947998285000878560311674, 9.007065153136923725670233193022, 9.875149719136042831238414980973, 11.10514374552009644547166763433, 11.75225051268908003480249341297

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