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Properties

Label 2-245-35.9-c3-0-22
Degree $2$
Conductor $245$
Sign $-0.330 + 0.943i$
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  + (−4.60 − 2.65i)2-s + (−1.67 + 0.969i)3-s + (10.1 + 17.5i)4-s + (−8.30 − 7.48i)5-s + 10.3·6-s − 65.0i·8-s + (−11.6 + 20.1i)9-s + (18.3 + 56.5i)10-s + (−12.7 − 22.1i)11-s + (−33.9 − 19.6i)12-s + 64.1i·13-s + (21.2 + 4.51i)15-s + (−91.7 + 158. i)16-s + (−23.9 + 13.8i)17-s + (106. − 61.7i)18-s + (0.396 − 0.686i)19-s + ⋯
L(s)  = 1  + (−1.62 − 0.939i)2-s + (−0.323 + 0.186i)3-s + (1.26 + 2.19i)4-s + (−0.742 − 0.669i)5-s + 0.701·6-s − 2.87i·8-s + (−0.430 + 0.745i)9-s + (0.580 + 1.78i)10-s + (−0.350 − 0.606i)11-s + (−0.817 − 0.472i)12-s + 1.36i·13-s + (0.365 + 0.0777i)15-s + (−1.43 + 2.48i)16-s + (−0.342 + 0.197i)17-s + (1.40 − 0.808i)18-s + (0.00478 − 0.00828i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.330 + 0.943i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.330 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-0.330 + 0.943i$
Analytic conductor: \(14.4554\)
Root analytic conductor: \(3.80203\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :3/2),\ -0.330 + 0.943i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.197764 - 0.278688i\)
\(L(\frac12)\) \(\approx\) \(0.197764 - 0.278688i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (8.30 + 7.48i)T \)
7 \( 1 \)
good2 \( 1 + (4.60 + 2.65i)T + (4 + 6.92i)T^{2} \)
3 \( 1 + (1.67 - 0.969i)T + (13.5 - 23.3i)T^{2} \)
11 \( 1 + (12.7 + 22.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 64.1iT - 2.19e3T^{2} \)
17 \( 1 + (23.9 - 13.8i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-0.396 + 0.686i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-94.0 - 54.3i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 234.T + 2.43e4T^{2} \)
31 \( 1 + (64.6 + 111. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (33.1 + 19.1i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 403.T + 6.89e4T^{2} \)
43 \( 1 + 172. iT - 7.95e4T^{2} \)
47 \( 1 + (179. + 103. i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (124. - 72.0i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (339. + 588. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-287. + 497. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-446. + 257. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 556.T + 3.57e5T^{2} \)
73 \( 1 + (-150. + 86.6i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-39.6 + 68.7i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.04e3iT - 5.71e5T^{2} \)
89 \( 1 + (-326. + 564. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 515. iT - 9.12e5T^{2} \)
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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

−11.33819663666233683626026483957, −10.49425079102131677985133875539, −9.359028431068550812892527212383, −8.571686739846369776689107399660, −7.932608524835148799364860082902, −6.74721569295722316019765756590, −4.85010561884017087014653232519, −3.41672068562194006736904696787, −1.90007675083968350933520202253, −0.36335210951995305417337764250, 0.806803722674075819338642346573, 2.88987450445783728431493019019, 5.16517667771442074822558017926, 6.43891957491093881983679512092, 7.04443232446903929074140969139, 8.030532374303876082523387612039, 8.757996050584935048203399037321, 10.01573034091373056912267778246, 10.64580493727576689335208909287, 11.53554007713713778968526756251

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