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Properties

Label 2-315-63.25-c1-0-4
Degree $2$
Conductor $315$
Sign $0.421 - 0.906i$
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  − 0.308·2-s + (−1.71 − 0.223i)3-s − 1.90·4-s + (0.5 − 0.866i)5-s + (0.529 + 0.0689i)6-s + (−2.36 − 1.17i)7-s + 1.20·8-s + (2.89 + 0.769i)9-s + (−0.154 + 0.266i)10-s + (0.550 + 0.953i)11-s + (3.27 + 0.426i)12-s + (2.54 + 4.40i)13-s + (0.729 + 0.362i)14-s + (−1.05 + 1.37i)15-s + 3.43·16-s + (−3.17 + 5.50i)17-s + ⋯
L(s)  = 1  − 0.217·2-s + (−0.991 − 0.129i)3-s − 0.952·4-s + (0.223 − 0.387i)5-s + (0.216 + 0.0281i)6-s + (−0.895 − 0.445i)7-s + 0.425·8-s + (0.966 + 0.256i)9-s + (−0.0487 + 0.0843i)10-s + (0.165 + 0.287i)11-s + (0.944 + 0.123i)12-s + (0.705 + 1.22i)13-s + (0.195 + 0.0969i)14-s + (−0.271 + 0.355i)15-s + 0.859·16-s + (−0.770 + 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.421 - 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.421 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.421 - 0.906i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.421 - 0.906i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.432736 + 0.276167i\)
\(L(\frac12)\) \(\approx\) \(0.432736 + 0.276167i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.71 + 0.223i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (2.36 + 1.17i)T \)
good2 \( 1 + 0.308T + 2T^{2} \)
11 \( 1 + (-0.550 - 0.953i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.54 - 4.40i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.17 - 5.50i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.518 - 0.897i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.186 + 0.322i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.91 - 3.31i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 + (-1.42 - 2.45i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.62 - 8.00i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.67 + 4.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 9.36T + 47T^{2} \)
53 \( 1 + (-2.35 + 4.08i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 4.66T + 59T^{2} \)
61 \( 1 + 8.30T + 61T^{2} \)
67 \( 1 - 2.33T + 67T^{2} \)
71 \( 1 + 2.37T + 71T^{2} \)
73 \( 1 + (5.94 - 10.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 10.5T + 79T^{2} \)
83 \( 1 + (7.01 - 12.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.40 + 5.90i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.00 + 3.47i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.88026238353049380203868759107, −10.76651977492364922761699039884, −9.946179138613893684487145761369, −9.195920146705619280839301068830, −8.144015538110752647346724888877, −6.71967438799343175047140285769, −6.06539663106983845879781843889, −4.66054445027947435362998569056, −3.95399966232493524915013641073, −1.33262806351514764194053406075, 0.52893217805447827376294878052, 3.10704699633660343091483810877, 4.49517648116437344631571910084, 5.62705737466020910405067734299, 6.36127456092354340189706730634, 7.62008806808912374270634195618, 8.940979546252431469800499870714, 9.694491244388839728877135140449, 10.46321100156164617153615184494, 11.39553635941312097571811869002

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