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Properties

Label 2-308-77.6-c1-0-6
Degree $2$
Conductor $308$
Sign $0.119 + 0.992i$
Analytic cond. $2.45939$
Root an. cond. $1.56824$
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  + (1.89 − 2.61i)3-s + (1.37 + 0.445i)5-s + (−1.79 − 1.94i)7-s + (−2.28 − 7.04i)9-s + (−0.0580 + 3.31i)11-s + (−0.296 − 0.911i)13-s + (3.76 − 2.73i)15-s + (−1.98 + 6.12i)17-s + (4.84 + 3.51i)19-s + (−8.48 + 0.981i)21-s + 5.57·23-s + (−2.36 − 1.71i)25-s + (−13.5 − 4.39i)27-s + (−3.22 − 4.44i)29-s + (6.30 − 2.04i)31-s + ⋯
L(s)  = 1  + (1.09 − 1.50i)3-s + (0.613 + 0.199i)5-s + (−0.676 − 0.736i)7-s + (−0.763 − 2.34i)9-s + (−0.0174 + 0.999i)11-s + (−0.0821 − 0.252i)13-s + (0.972 − 0.706i)15-s + (−0.482 + 1.48i)17-s + (1.11 + 0.807i)19-s + (−1.85 + 0.214i)21-s + 1.16·23-s + (−0.472 − 0.343i)25-s + (−2.60 − 0.846i)27-s + (−0.599 − 0.824i)29-s + (1.13 − 0.367i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.119 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.119 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $0.119 + 0.992i$
Analytic conductor: \(2.45939\)
Root analytic conductor: \(1.56824\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :1/2),\ 0.119 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37009 - 1.21522i\)
\(L(\frac12)\) \(\approx\) \(1.37009 - 1.21522i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (1.79 + 1.94i)T \)
11 \( 1 + (0.0580 - 3.31i)T \)
good3 \( 1 + (-1.89 + 2.61i)T + (-0.927 - 2.85i)T^{2} \)
5 \( 1 + (-1.37 - 0.445i)T + (4.04 + 2.93i)T^{2} \)
13 \( 1 + (0.296 + 0.911i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.98 - 6.12i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-4.84 - 3.51i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 5.57T + 23T^{2} \)
29 \( 1 + (3.22 + 4.44i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-6.30 + 2.04i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (5.93 - 4.31i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-2.54 - 1.84i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 0.0165iT - 43T^{2} \)
47 \( 1 + (-3.77 + 5.20i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.478 + 1.47i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-6.37 - 8.78i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.00 + 3.10i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 5.45T + 67T^{2} \)
71 \( 1 + (-0.241 + 0.743i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (6.71 - 4.87i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (6.11 - 1.98i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.64 - 8.14i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 2.07iT - 89T^{2} \)
97 \( 1 + (4.98 - 1.61i)T + (78.4 - 57.0i)T^{2} \)
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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

−11.84520635144679510418562564531, −10.25330192595774229038813569085, −9.593611698918662560509762640583, −8.444438728522472688792050672242, −7.53832675686999091181327019133, −6.82037813326460818681733728287, −5.95800353241129887770779452932, −3.87556627157751332727531782218, −2.64276880444192163389055383759, −1.44252274401088908056928481933, 2.65665561246624585167751007399, 3.32677363695495805364951672707, 4.86120707051613298177283764495, 5.58628497321885190629593139581, 7.20814210591422987392426755044, 8.719735158569164128258718351766, 9.167839833552707602456232091602, 9.671274978336267662598983490733, 10.79497342436489997229185379071, 11.65342147054763492089547896441

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