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Properties

Label 2-315-63.38-c1-0-0
Degree $2$
Conductor $315$
Sign $0.204 + 0.978i$
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  + 2.32i·2-s + (−1.01 + 1.40i)3-s − 3.40·4-s + (−0.5 + 0.866i)5-s + (−3.26 − 2.35i)6-s + (−1.17 − 2.37i)7-s − 3.26i·8-s + (−0.941 − 2.84i)9-s + (−2.01 − 1.16i)10-s + (−3.50 + 2.02i)11-s + (3.45 − 4.77i)12-s + (3.52 − 2.03i)13-s + (5.51 − 2.72i)14-s + (−0.708 − 1.58i)15-s + 0.775·16-s + (−2.04 + 3.54i)17-s + ⋯
L(s)  = 1  + 1.64i·2-s + (−0.585 + 0.810i)3-s − 1.70·4-s + (−0.223 + 0.387i)5-s + (−1.33 − 0.962i)6-s + (−0.442 − 0.896i)7-s − 1.15i·8-s + (−0.313 − 0.949i)9-s + (−0.636 − 0.367i)10-s + (−1.05 + 0.610i)11-s + (0.996 − 1.37i)12-s + (0.978 − 0.564i)13-s + (1.47 − 0.727i)14-s + (−0.182 − 0.408i)15-s + 0.193·16-s + (−0.496 + 0.860i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.204 + 0.978i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.204 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.297253 - 0.241534i\)
\(L(\frac12)\) \(\approx\) \(0.297253 - 0.241534i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.01 - 1.40i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (1.17 + 2.37i)T \)
good2 \( 1 - 2.32iT - 2T^{2} \)
11 \( 1 + (3.50 - 2.02i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.52 + 2.03i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.04 - 3.54i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.39 - 2.54i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.66 - 2.69i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.66 + 3.84i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.74iT - 31T^{2} \)
37 \( 1 + (3.38 + 5.86i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.50 - 4.34i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.96 - 6.86i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + (-0.306 - 0.177i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 5.67T + 59T^{2} \)
61 \( 1 + 1.52iT - 61T^{2} \)
67 \( 1 + 5.76T + 67T^{2} \)
71 \( 1 + 8.87iT - 71T^{2} \)
73 \( 1 + (-5.36 - 3.09i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 9.47T + 79T^{2} \)
83 \( 1 + (7.28 - 12.6i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.57 - 6.18i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.13 - 4.69i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.75539453969265602612629421738, −11.00150690226236478443110403966, −10.55924790441522275140802454122, −9.475433484081214738170711816291, −8.355084659547709646607219680835, −7.44973870375004295202082232512, −6.47999088942366657366678047338, −5.73131405688320917221309588604, −4.59150636464246654514845169487, −3.62075118615328246800716742480, 0.28714411797832093323073080391, 2.00388510129389767987372261759, 3.05854659794037912310255806012, 4.65959139318288361649337692629, 5.72839486835287945342133353199, 6.98578997776082391412886179802, 8.590299639800176584354283062644, 9.004510898198513719476491748122, 10.45522184498174182023641981186, 11.20702371697642370715456429032

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