Properties

Label 2-245-245.103-c3-0-15
Degree $2$
Conductor $245$
Sign $-0.972 - 0.230i$
Analytic cond. $14.4554$
Root an. cond. $3.80203$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.486 + 0.658i)2-s + (−0.458 + 0.0866i)3-s + (2.16 + 7.00i)4-s + (9.36 − 6.11i)5-s + (0.165 − 0.343i)6-s + (−9.16 − 16.0i)7-s + (−11.8 − 4.14i)8-s + (−24.9 + 9.78i)9-s + (−0.526 + 9.13i)10-s + (−17.1 + 43.6i)11-s + (−1.59 − 3.02i)12-s + (−0.134 − 0.0151i)13-s + (15.0 + 1.78i)14-s + (−3.75 + 3.61i)15-s + (−39.9 + 27.2i)16-s + (6.40 + 0.239i)17-s + ⋯
L(s)  = 1  + (−0.171 + 0.232i)2-s + (−0.0881 + 0.0166i)3-s + (0.270 + 0.875i)4-s + (0.837 − 0.546i)5-s + (0.0112 − 0.0233i)6-s + (−0.494 − 0.868i)7-s + (−0.523 − 0.183i)8-s + (−0.923 + 0.362i)9-s + (−0.0166 + 0.288i)10-s + (−0.469 + 1.19i)11-s + (−0.0384 − 0.0726i)12-s + (−0.00286 − 0.000323i)13-s + (0.287 + 0.0341i)14-s + (−0.0647 + 0.0621i)15-s + (−0.624 + 0.425i)16-s + (0.0913 + 0.00341i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.230i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.972 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-0.972 - 0.230i$
Analytic conductor: \(14.4554\)
Root analytic conductor: \(3.80203\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :3/2),\ -0.972 - 0.230i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0773188 + 0.660450i\)
\(L(\frac12)\) \(\approx\) \(0.0773188 + 0.660450i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-9.36 + 6.11i)T \)
7 \( 1 + (9.16 + 16.0i)T \)
good2 \( 1 + (0.486 - 0.658i)T + (-2.35 - 7.64i)T^{2} \)
3 \( 1 + (0.458 - 0.0866i)T + (25.1 - 9.86i)T^{2} \)
11 \( 1 + (17.1 - 43.6i)T + (-975. - 905. i)T^{2} \)
13 \( 1 + (0.134 + 0.0151i)T + (2.14e3 + 488. i)T^{2} \)
17 \( 1 + (-6.40 - 0.239i)T + (4.89e3 + 367. i)T^{2} \)
19 \( 1 + (26.6 - 46.0i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-3.11 - 83.3i)T + (-1.21e4 + 909. i)T^{2} \)
29 \( 1 + (279. - 63.7i)T + (2.19e4 - 1.05e4i)T^{2} \)
31 \( 1 + (-45.5 + 26.3i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (319. - 169. i)T + (2.85e4 - 4.18e4i)T^{2} \)
41 \( 1 + (-30.4 - 63.1i)T + (-4.29e4 + 5.38e4i)T^{2} \)
43 \( 1 + (-67.4 - 192. i)T + (-6.21e4 + 4.95e4i)T^{2} \)
47 \( 1 + (-23.6 - 17.4i)T + (3.06e4 + 9.92e4i)T^{2} \)
53 \( 1 + (-5.44 - 2.87i)T + (8.38e4 + 1.23e5i)T^{2} \)
59 \( 1 + (41.0 + 547. i)T + (-2.03e5 + 3.06e4i)T^{2} \)
61 \( 1 + (-114. + 371. i)T + (-1.87e5 - 1.27e5i)T^{2} \)
67 \( 1 + (78.4 + 21.0i)T + (2.60e5 + 1.50e5i)T^{2} \)
71 \( 1 + (-79.7 + 349. i)T + (-3.22e5 - 1.55e5i)T^{2} \)
73 \( 1 + (463. - 342. i)T + (1.14e5 - 3.71e5i)T^{2} \)
79 \( 1 + (-193. - 111. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (127. + 1.13e3i)T + (-5.57e5 + 1.27e5i)T^{2} \)
89 \( 1 + (-475. - 1.21e3i)T + (-5.16e5 + 4.79e5i)T^{2} \)
97 \( 1 + (-900. - 900. i)T + 9.12e5iT^{2} \)
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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

−12.28134686504663927707324493753, −11.10886990293070974922683795822, −10.04881354665250948838144593405, −9.198991089467897066604544454889, −8.072905901744149450897535900515, −7.22759707167230369130939789856, −6.11458660784417797356849437067, −4.88732893823854772478811176639, −3.45494747285121030631960644478, −2.00448370220077102734081208162, 0.25420466500125947959507916051, 2.21095335032526128630416611906, 3.11019911989082274512790204028, 5.58658705841622575074885548336, 5.79482027610585978719643532439, 6.85629523465621767618650423284, 8.712415496095383306023922570120, 9.236532457539133635226937375126, 10.39196589413057176539367524002, 11.04076798223907582130084327527

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