Properties

Label 2-245-35.4-c3-0-10
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: $\chi_{245} (214, \cdot )$
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

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

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